ProfessorStewart’sCasebookofMathematicalMysteries

ProfessorIanStewartisknownthroughouttheworldformakingmathematicspopular.HereceivedtheRoyalSociety’sFaradayMedalforfurtheringthepublicunderstandingofsciencein1995,theIMAGoldMedalin2000,theAAASPublicUnderstandingofScienceandTechnologyAwardin2001andtheLMS/IMAZeemanMedalin2008.HewaselectedaFellowoftheRoyalSocietyin2001.HeisEmeritusProfessorofMathematicsattheUniversityofWarwick,wherehedivideshistimebetweenresearchintononlineardynamicsandfurtheringpublicawarenessofmathematics.Hismanypopularsciencebooksinclude(withTerryPratchettandJackCohen)TheScienceofDiscworldItoIV,TheMathematicsofLife,17EquationsthatChangedtheWorldandTheGreatMathematicalProblems.Hisapp,ProfessorStewart’sIncredibleNumbers,waspublishedjointlybyProfileandTouchPressinMarch2014.

BytheSameAuthorConceptsOfModernMathematicsGame,Set,AndMathDoesGodPlayDice?AnotherFineMathYou’veGotMeIntoFearfulSymmetryNature’sNumbersFromHereToInfinityTheMagicalMazeLife’sOtherSecretFlatterlandWhatShapeIsASnowflake?TheAnnotatedFlatlandMathHysteriaTheMayorOfUglyville’sDilemmaHowToCutACakeLettersToAYoungMathematicianTamingTheInfinite(AlternativeTitle:TheStoryOfMathematics)WhyBeautyIsTruthCowsInTheMazeMathematicsOfLifeProfessorStewart’sCabinetOfMathematicalCuriositiesProfessorStewart’sHoardOfMathematicalTreasuresSeventeenEquationsThatChangedTheWorld(AlternativeTitle:InPursuitOfTheUnknown)TheGreatMathematicalProblems(AlternativeTitle:VisionsOfInfinity)Symmetry:AVeryShortIntroductionJackOfAllTrades(ScienceFictioneBook)

withJackCohenTheCollapseOfChaosEvolvingTheAlien(AlternativeTitle:WhatDoesAMartianLookLike?)FigmentsOfRealityWheelers(ScienceFiction)Heaven(ScienceFiction)

TheScienceOfDiscworldSeries(WithTerryPratchett&JackCohen)TheScienceOfDiscworldTheScienceOfDiscworldII:TheGlobeTheScienceOfDiscworldIII:Darwin’sWatchTheScienceOfDiscworldIV:JudgementDay

iPadappIncredibleNumbers

ProfessorStewart’sCasebookofMathematicalMysteries

IanStewart

Firstpublishedin2014byPROFILEBOOKSLTD3AExmouthHousePineStreetLondonEC1R0JHwww.profilebooks.com

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eISBN9781847654328

CONTENTS

Acknowledgements

IntroducingSoamesandWatsupNoteonUnitsTheScandaloftheStolenSovereignNumberCuriosityTrackPositionSoamesMeetsWatsupGeomagicSquaresWhatShapeisanOrangePeel?HowtoWintheLottery?TheGreenSocksCaperIncidentConsecutiveCubesAdonisAsteroidMousterianTwoSquareQuickiesCaughtClean-HandedTheAdventureoftheCardboardBoxesTheRATSSequenceBirthdaysAreGoodforYouMathematicalDatesTheHoundoftheBasketballsDigitalCubesNarcissisticNumbersPiphilology,Piems,andPilishClueless!ABriefHistoryofSudokuHexakosioihexekontahexaphobiaOnce,Twice,ThriceConservationofLuckTheCaseoftheFace-DownAcesConfusedParents

JigsawParadoxTheCatflapofFearPancakeNumbersTheSoupPlateTrickMathematicalHaikuTheCaseoftheCrypticCartwheelTwobyTwoTheV-shapedGooseMysteryEelishMnemonicsAmazingSquaresTheThirty-SevenMysteryAverageSpeedFourCluelessPseudokuSumsofCubesThePuzzleofthePurloinedPapersMasterofAllHeSurveysAnotherNumberCuriosityTheOpaqueSquareProblemOpaquePolygonsandCirclesπr2?TheSignofOneProgressonPrimeGapsTheOddGoldbachConjecturePrimeNumberMysteriesTheOptimalPyramidTheSignofOne:PartTwoInitialConfusionEuclid’sDoodleEuclideanEfficiency123456789TimesXTheSignofOne:PartThreeTaxicabNumbersTheWaveofTranslationRiddleoftheSandsEskimoπTheSignofOne:PartFour—ConcludedSeriouslyDerangedTossingaFairCoinIsn’tFair

PlayingPokerbyPostEliminatingtheImpossibleMusselPowerProofThattheWorldisRound123456789TimesXContinuedThePriceofFameTheRiddleoftheGoldenRhombusAPowerfulArithmeticSequenceWhyDoGuinnessBubblesGoDownwards?RandomHarmonicSeriesTheDogsThatFightintheParkHowTallisThatTree?WhyDoMyFriendsHaveMoreFriendsThanIDo?Isn’tStatisticsWonderful?TheAdventureoftheSixGuestsHowtoWriteVeryBigNumbersGraham’sNumberCan’tWrapMyHeadAroundItTheAffairoftheAbove-AverageDriverTheMousetrapCubeSierpińskiNumbersJamesJosephWho?TheBafflehamBurglaryTheQuadrillionthDigitofπIsπNormal?AMathematician,aStatistician,andanEngineer…LakesofWadaFermat’sLastLimerickMalfatti’sMistakeSquareLeftoversCoinTossingoverthePhoneHowtoStopUnwantedEchoesTheEnigmaoftheVersatileTileTheThrackleConjectureBargainwiththeDevilATilingThatIsNotPeriodicTheTwoColourTheoremTheFourColourTheoreminSpace

ComicalCalculusTheErdősDiscrepancyProblemTheGreekIntegratorSumsofFourCubesWhytheLeopardGotItsSpotsPolygonsForeverTopSecretTheAdventureoftheRowingMenTheFifteenPuzzleTheTrickySixPuzzleAsDifficultasABCRingsofRegularSolidsTheSquarePegProblemTheImpossibleRouteTheFinalProblemTheReturnTheFinalSolution

TheMysteriesDemystified

Acknowledgements

page13Leftandcentrefigures:LaurentBartholdiandAndréHenriques.OrangepeelsandFresnelintegrals,MathematicalIntelligencer34No.4(2012)1–3.

page13Rightfigure:LucDevroye.page27Boxpuzzleconcept:MoloyDe.page40ExtractfromNotaWake:MikeKeith.page62Haiku:DanielMathews,JonathanAlperin,JonathanRosenberg.page67Figure:http://getyournotes.blogspot.co.uk/2011/08/why-do-some-birds-

fly-in-v-formations.htmlpage71Amazingsquares:devisedbyMoloyDeandNirmalyaChattopadhyay.page72TheThirty-SevenMystery:basedonobservationsbyStephenGledhill.page74Cluelesspseudoku:GerardButters,FrederickHenle,JamesHenleand

ColleenMcGaughey.Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.

page96Figure:EricW.Weisstein,‘Brocard’sConjecture,’fromMathWorld—AWolframWebResource:http://mathworld.wolfram.com/BrocardsConjecture.html

page101RightFigure:StevenSnape.page109Figure:CourtesyoftheUW-MadisonArchives.page123Leftfigure:[GeorgeSteinmetz,courtesyofAnastasiaPhoto].page123Rightfigure:NASA,HiRISEonMarsReconnaissanceOrbiter.page124Rightfigure:RudiPodgornik.page125Figure:VeitSchwa¨mmleandHansJ.Herrmann.Solitarywave

behaviourofsanddunes,Nature426(2003)619–620.page131Figure:PersiDiaconis,SusanHolmesandRichardMontgomery,

Dynamicalbiasinthecointoss,SIAMReview49(2007)211–223.page205Figures:JoshuaSocolarandJoanTaylor.Anaperiodichexagonaltile,JournalofCombinatorialTheorySeriesA118(2011)2207–2231;

http://link.springer.com/article/10.1007%2Fs00283-011-9255-ypages235–7Figures:MichaelElgersmaandStanWagon,ClosingaPlatonic

gap,TheMathematicalIntelligencer(2014)toappear.

ThefollowingfiguresarereproducedundertheCreativeCommonsAttribution3.0Unportedlicenseandcreditedasrequestedonthedescriptionpage:page95Krishnavedala.page101(left)RicardoLiberato.page102Tekisch.page139AndreasTrepte,www.photo-natur.de.page180Braindrain0000.page183LutzL.page296WaltersArtMuseum,Baltimore.

IntroducingSoamesandWatsup

ProfessorStewart’sCabinetofMathematicalCuriositiesappearedin2008,justbeforeChristmas.Readersseemedtolikeitsrandommixtureofquirkymathematicaltricks,games,weirdbiographies,snippetsofstrangeinformation,solvedandunsolvedproblems,oddfactoids,andtheoccasionallongerandmoreseriouspieceontopicssuchasfractals,topology,andFermat’sLastTheorem.Soin2009itwasfollowedbyProfessorStewart’sHoardofMathematicalTreasures,whichcontinuedinthesameveinwithanintermittentpiratetheme.

Theysaythatthreeisagoodnumberforatrilogy.ThelateDouglasAdamsofTheHitchhiker’sGuidetotheGalaxyfamedideventuallydecidethatfourwasbetterandfivebetterstill,butthreesoundslikeagoodplacetostart.So,afteragapoffiveyears,hereisProfessorStewart’sCasebookofMathematicalMysteries.Thistime,however,there’sanewtwist.Theshortquirkyitems,suchasHexakosioihexekontahexaphobia,theThrackleConjecture,WhatShapeisanOrangePeel?,theRATSSequence,andEuclid’sDoodle,arestillthere.Soaremoresubstantialarticlesaboutsolvedandunsolvedproblems:PancakeNumbers,theGoldbachConjecture,theErdősDiscrepancyProblem,theSquarePegConjecture,andtheABCConjecture.Soarethejokes,poems,andanecdotes.Nottomentionunusualapplicationsofmathematicstoflyinggeese,clumpsofmussels,spottyleopards,andbubblesinGuinness.ButthesemiscellaneaarenowinterspersedwithaseriesofnarrativeepisodesfeaturingaVictoriandetectiveandhismedicalsidekick—

Iknowwhatyou’rethinking.However,IdevelopedtheideaayearorsobeforeBenedictCumberbatchandMartinFreeman’sspectacularlysuccessfulmoderntakeonSirArthurConanDoyle’smuch-lovedcharactershitthesmallscreen.(Trustme.)Moretothepoint,it’snotthatpair.NotevenasportrayedinSirArthur’soriginalstories.Yes,myguysliveintheoriginaltimeperiod,butacrosstheroadatnumber222B.Fromthere,theycastenviouseyesonthestreamofrichclientsenteringthepremisesofthemorefamousduo.Andfromtimetotimeacasecomesupthattheirillustriousneighbourshaveshunnedorfailedtosolve:sucharcanemysteriesastheSignofOne,theDogsthatFightinthePark,theCatflapofFear,andtheGreekIntegrator.ThenHemlockSoames

andDrJohnWatsupputtheirbrainsingear,showtheirtruecoloursandtheirstrengthofcharacter,andtriumphoveradversityandlackofmarketpresence.

Thesearemathematicalmysteries,youappreciate.Theirsolutionsdemandaninterestinmathematicsandanabilitytothinkclearly,attributesinwhichSoamesandWatsuparebynomeansdeficient.Thesepassagesaresignalledbythesymbol .AlongthewaywelearnofWatsup’spriormilitarycareerinAl-JebraistanandSoames’sbattleswithhisarch-enemyProfessorMogiarty,inevitablyleadingtothefinalfatalconfrontationatoptheSchtickelbachFalls.Andthen—

ItisfortunatethatDrWatsuprecordedsomanyoftheirjointinvestigationsinhismemoirsandunpublishednotes.IamgratefultohisdescendantsUnderwoodandVerityWatsupforpermittingmeunprecedentedaccesstofamilydocuments,andforgenerouslygrantingmepermissiontoincludeextractshere.

Coventry,March2014

NoteonUnits

InSoamesandWatsup’sera,thestandardunitsofmeasurementinBritainwereimperial,notmetricastheymostlyaretoday,andthecurrencywasnotdecimal.Americanreaderswillhavenoproblemswithimperialunits;admittedly,thegallonisdifferentoneithersideoftheAtlantic,butthatunitofmeasurementdoesn’tappearanyway.ToavoidinconsistenciesI’veusedunitsappropriatetotheVictorianera,evenfortopicsthatarenotpartoftheSoames/Watsupcanon,exceptwhennarrativeimperativedemandsmetric.

Here’saquickguidetotherelevantunitswithmetric/decimalequivalents.Mostofthetimetheactualunitdoesn’tmatter:youcouldleavethenumbers

unchanged,butcrossout‘inch’or‘yard’andreplaceitbyanunspecified‘unit’.Orchoosewhicheverseemsconvenient(metreforyard,forexample).Lengths

1foot(ft)=12inches(in) 304·8mm1yard(yd)=3feet 0·9144m1mile(mi)=1760yards=5040feet 1·609km1league(lea)=3miles 4·827kmWeights

1pound(lb)=16ounces(oz) 453·6g1stone(st)=14pounds 6·35kg1hundredweight(cwt)=8stone=112pounds 50·8kg1ton(t)=20hundredweights=2240pounds 1·016tonnesMoney

1shilling(s)=12pence(d)[singular:penny] 5newpence1pound(£)=20shillings=240pence1sovereign=1pound(coin)

1guinea=£1.1s. £1.051crown=5s. 25newpenceThruppenybit=colloquialtermforathreepencecoin.

TheScandaloftheStolenSovereignTheprivatedetectivetookhiswalletfromhispocket,ascertainedthatitwasstillempty,andsighed.Standingatthewindowofhislodgingsatnumber222Bhestaredmoroselyacrossthestreet.ThestrainsofanIrishair,expertlyplayedonaStradivarius,werejustdiscernibleabovetheclatterofpassingcarriages.Really,themanwasinsufferable!Soamesstaredatthestreamofpeopleenteringtheportalsofhisfamouscompetitor.Mostwerewealthymembersoftheupperclasses.Thosethatappearednottobewealthymembersoftheupperclasseswere,withfewexceptions,representingwealthymembersoftheupperclasses.

Criminalsjustweren’tcommittingthekindofcrimethataffectedthesortofpeoplewhowouldengagetheservicesofHemlockSoames.

Forthepasttwoweeks,Soameshadwatchedwithenviouseyesasclientafterclientwasusheredintothepresenceofthepersontheybelievedtobethegreatestdetectiveintheworld.Or,atleast,inLondon,whichforVictorianEnglandamountedtothesamething.Meanwhilehisowndoorbellremainedmute,thebillspiledup,andMrsSoapsudswasthreateningeviction.

Hehadonlyonecaseonthebooks.LordHumphshaw-Smattering,owneroftheGlitzHotel,believedthatoneofhiswaitershadstolenagoldsovereign:value,onepoundsterling.Tobefair,Soamescoulddowithasovereignhimselfrightnow.Butitwashardlythestufftoattractthesensationalistyellowpress,uponwhom,deplorableasitmightbe,hisfuturedepended.

Soamesstudiedhiscasenotes.Threefriends,Armstrong,Bennett,andCunningham,hadpartakenofdinneratthehotel,andhadbeenpresentedwithabillfor£30.EachhadgiventhewaiterManueltengoldsovereigns.Butthenthemaîtred’noticedthattherehadbeenanerror,andthebillwasactually£25.Hegavethewaiterfivesovereignstoreturntothemen.Since£5wasn’tdivisibleby3,Manuelsuggestedthathemightkeeptwoofthecoinsasatipandgivethembackonesovereigneach,hintingthattheywerefortunatetohaveanyoftheoverpaymentreturned.

Thecustomersagreed,andallwaswelluntilthemaîtred’noticedanarithmeticaldiscrepancy.Nowthemenhadeachpaid£9,atotalof£27.Manuelhadafurther£2,makingatotalof£29.

Onepoundwasmissing.Humphshaw-SmatteringwasconvincedthatManuelhadstolenit.Although

theevidencewascircumstantial,Soamesknewthatthewaiter’slivelihood

dependedonresolvingthemystery.IfManuelweretobedismissedwithabadreference,hewouldnevergetanotherjob.

Wheredidthemissingpoundgo?

Seepage249fortheanswer.

NumberCuriosity*

Indetectivework,itisvitaltobeabletospotapattern.Soames’sunpublishedanduntitledmonographcontainingtwothousandandforty-oneinstructiveexamplesofpatternsincludesthefollowing.Workout

11×9111×909111×90909111×9090909111×9090909091

Soameswouldhaveusedpenandpaper,andmodernreadersmaydolikewiseiftheycanrememberhow.Acalculatorisalwaysanoption,buttheytendtorunoutofdigits.Thepatterncontinuesindefinitely:thiscan’tbeprovedusingacalculator,butitcanbededucedfromtheold-fashionedmethod.So,withoutdoinganyfurthercalculations,whatis

11×9090909090909091?

Aharderquestionis:whydoesitwork?

Seepage250fortheanswers.

Footnote

*Manyitemsinthiscompilationthatdonotreferdirectlytocriminalcasesareextractedfromhandwrittennotes,someofwhosecontentshavebeencollectedandpublished,withSoames’spermission,asDoctorWatsup’sVaultofForensicAnomalies,andwillbereproducedwithoutfurthernotification.Someareoflaterdate,addedbyWatsup’sliteraryexecutors,andtheassiduousreaderwillinstantlyidentifysuchanachronisms.

TrackPositionLionelPenroseinventedavariationontraditionalmazes:railwaymazes.Thesehavejunctionslikethoseonrailwaytracks,andyouhavetotakearoutethroughthemthatatraincouldfollow,onewithnosharpturns.Theyareagoodwaytocramacomplicatedmazeintoasmallspace.

Allowedandforbiddenroutesatjunctions

Hisson,themathematicianRogerPenrose,tooktheideafurther.OneofhismazesiscarvedinstoneontheLuppittMillenniumBench,inDevon,England.Thatone’sabitdifficult,sohere’sasimplerexampleforyoutotackle.

ThemapoverleafshowstherailnetworkofTardyTrains.The10.33trainstartsatstationSandmustfinishatstationF.Thetraincannotreversedirectionbyslowingdownandthengoingbackwards,butitcantravelalongalineineitherdirectionifthetrackloopsbackonitself.Atpoints,wheretwobranchesjoin,thetrainmaytakeanysmoothpath.Whatroutedoesthetraintake?

Themaze

Seepage252fortheanswer,andfurtherinformationincludingtheLuppittMillenniumMaze.

SoamesMeetsWatsupAfinedrizzle,ofthekindthatlooksinnocuousbutquicklysoaksyoutotheskin,wasfallingonthegoodcitizensofLondon,andonthebad,astheyscurriedalongBakerStreetonerrandsadmirableornefarious,dodgingthepuddles.Thenot-so-famousdetectivewasinhishabitualpositionatthewindow,staringhopelesslyintothegloom,grumblingtohimselfabouthisdirefinances,andfeelingdepressed.HisincisivesolutiontotheScandaloftheStolenSovereignhadbroughtinenoughtogetMrsSoapsudstemporarilyoffhisback,butnowthattheemotionalrushofsuccesshadsubsided,hefeltlonelyandunappreciated.

Perhapsheneededalike-mindedcompanion?Onewhocouldsharethedailycut-and-thrustofhispersonalvendettaagainstcrime,andtheintellectualchallengeofunravellingthecluesthatitsperpetratorsscatteredsocarelesslyacrossthelandscape?Butwherecouldhefindsuchaperson?Hehadnoideawheretostart.

Hisblackmoodwasinterruptedbytheappearanceofasturdyfigurestridingpurposefullytowardsthepremisesopposite.Instinctively,Soamesjudgedhimtobeamedicalman,recentlyretiredfromthearmy.Well-dressed,well-heeled:yetanotherwealthyclientforthatoverratedjackassHol—

Butno!Thefigureinspectedthehousenumber,shookhishead,andspunonhisheels.Ashecrossedtheroad,narrowlydodgingahansomcab,thebrimofhishatconcealedhisface,buthisbodylanguageshoweddetermination,perhapsvergingondesperation.Observingthemanmoreclosely,nowthathisinteresthadbeenpiqued,Soamesrealisedthathiscoatwasnotnew,ashehadfirstthought.Ithadbeenexpertlyrepaired…inOldComptonStreet,bythelookofthestitching.OnaThursday,whentheheadseamstressestookahalf-dayoff.Downatheel,notwell-heeled,hecorrectedhisinitialimpression,asthemandisappearedfromview,apparentlyheadingforthedoorwaybelow.

Apause:thenthebellrang.Soameswaited.Aknockatthedoorannouncedhislong-sufferinglandlady

MrsSoapsuds,cladinoneofherhabitualfloralprintdressesandwearingalargepinafore.“Agentlemantoseeyou,MrSoames,”shesniffed.“ShallIshowhimup?”

Soamesnodded,andMrsSoapsudsslouchedoffdownthestairs.Aminute

latersheknockedagain,andthemedicalgentlemanentered.Soameswavedathertoshutthedoorandreturntohercustomaryplacebehindthenetcurtainsinhersitting-roomonthegroundfloor,whichshedidwithevidentreluctance.

Thegentlemanlistenedforamoment,andsuddenlytuggedthedooropen,steppingbacktoallowMrsSoapsudstofallsidewaystothefloor.

“The—uh—mat.Neededdusting,”sheexplained,pickingherselfup.Soamessilentlynotedthathislandladyalsoneededdusting,gaveherathinsmile,andwavedheraway.Oncemorethedoorclosed.

“Mycard,”themansaid.Soamesplacedthevisitingcardfacedown,unread,andstudiedthenew

arrivalfromheadtotoe.Afterafewsecondshesaid,“Notmuchofnotetoidentifyyou.”

“Pardon?”“Excepttheobvious,ofcourse.YouhavebeeninAl-Jebraistanforthelast

fouryears,servingasasurgeonwiththeRoyalSixthDragoons.YounarrowlyescapedaseriouswoundattheBattleofQ’drat.Yourperiodofserviceendedsoonafter,andyoudecided—aftersomesoul-searching—toreturntoEngland,whichyoudidearlythisyear.”Soamespeeredmoreclosely,andadded,“Youkeepfourcats.”

Astheman’sjawdropped,Soamesturnedoverthecard.“DrJohnWatsup,”heread.“Surgeon,RoyalSixthDragoons,retired.”Hisfaceshowednoemotionatthisconfirmationofhisdeductions,forithadbeeninevitable.“Pleasesitdown,sir,andtellmeofthecrimethathasbeencommittedagainstyou.Icanassureyouthat—”

Watsuplaughed,afriendlychuckle.“MrSoames,Iamdelightedtohavemetyouatlast,foryourfamehasspreadfarandwide.YourdeductionsaboutmypersonprovethatyoufullydeservetheacclaimthatIhaveencountered.Yourmodestyatthefeatbecomesyou.ButIdonotcomeprimarilyasaprospectiveclient.Rather,Iamseekingapositioninyouremploy.Medicinenolongerappealstome—norwouldittoyouifyouhadseenthesightsIhavebeenforcedtoendureatthebattlefront.ButIamamanofaction,Icontinuetocraveexcitement,Istillhavemyservicerevolver,and…bytheway,howdidyoudothat?”

Soames,ignoringagrowingfeelingthathewasbeingmistakenfortheinhabitantofnumber221B,satdownfacingWatsup.“Byyourbearing,sir,Ihadyoumarkedasamilitarymanbeforeyoucrossedtheroad.Myeyesightispreternaturallykeen,andyouhavethehandsofasurgeon,strongyetlackingtheingrainedstainsofmanuallabour.LastDecembertheTimesreportedthatthefour-yearcampaigninAl-JebraistanwascomingtoacloseandtheRoyalSixth

DragoonswerereturningtoEnglandafterfightingadecisivebutcostlybattleatQ’drat.Youarewearingtheappropriateregimentalboots,andthewear-patternsonthemshowyouhavebeenbackinEnglandforsometime.Youhaveaslightscaralongyourjawbone,almosthealed,whichwasobviouslycausedbyamusket-ballofnon-Europeandesign—IhavewrittenabriefmonographonfirearminjuriesintheFarEast,Imustreadittoyousometime.Youareamanofaction,asevincedbythewayyouhandledMrsSoapsuds’ssnooping,soyouwouldnothaveretiredfrommilitaryservicevoluntarily.IfyouhadbeengivenadishonourabledischargeIwouldhaveseenitreportedinthescandalsheets,butnothingofthekindhasbeenpublishedrecently.Yourcoatbearsfourdifferenttypesofcathair—notjustfourcolours,whichmightindicateasingletabby,butdifferentlengthsandtextures…Iwillspareyoualistoftheirbreeds.”

“Astonishing!”“Tobecandid,Imustalsoadmitthatyourfaceisfamiliar.Iamsurethat

somewhere—ah,yes!Ihaveit!Asmallarticleinlastweek’sChronicle,withaphotograph…DrJohnWatsup,originatorofthewell-knownphrase‘Watsup,doc?’Yourfameexceedsmyown,Doctor.”

“Youaretookind,MrSoames.”“No,merelyrealistic.Butifwearetoworktogether,youmustconvinceme

thatyoucanthinkaswellasact.Letussee.”AndSoameswrotethedigits

49

onthebackofanenvelope.“Iwantyoutoinsertonestandardarithmeticalsymbol,toproduceawholenumberbetween1and9.”

Watsuppursedhislipsinconcentration.“Aplussign…no,13istoolarge.Aminus—notheresultisnegative.Neithermultiplicationnordivisionwilldo.Ofcourse!Asquareroot!Oh,no: ,againtoolarge.”Hescratchedhishead.“Iamstumped.Itisimpossible.”

“Iassureyouthereisasolution.”Thesilencewasbrokenonlybythetickingofaclockonthemantelpiece.

Suddenly,Watsup’sfacelitup.“Ihaveit!”Hepickeduptheenvelope,addedasinglesymbol,andhandedittoSoames.

“Youpassthefirsttest,Doctor.”

WhatdidWatsupwrite?Seepage252fortheanswer.

GeomagicSquaresAmagicsquareismadefromnumbers,whichgivethesametotalalonganyrow,column,ordiagonal.LeeSallowshasinventedageometricanalogue,thegeomagicsquare.Thisisasquarearrayofshapes,suchthattheshapesinanyrow,column,ordiagonalfittogetherlikeajigsawtomakethesameoverallshape.Thepiecescanberotatedorreflectedifnecessary.Theleft-handfigureshowshowthisgoes;theright-handoneisapuzzleforyoutosolve.Seepage253fortheanswer.

Sallowshasinventedmanyothergeomagicsquares,alongwithgeneralisationssuchasageomagictriangle.SeeTheMathematicalIntelligencer33No.4(2011)25–31andhiswebsite

http://www.GeomagicSquares.com/

TwoofLeeSallows’sgeomagicsquares.Followarow,column,ordiagonaltofindtheassembledjigsawusingthecorrespondingpieces.Left:Acompletedexample.Right:Yourtaskisfindtheassembledjigsaws

forallrows,columns,anddiagonals.

WhatShapeisanOrangePeel?Therearemanywaystopeelanorange.Someofusbreakbitsoff.Sometrytoremovetheentirepeelinasingleirregularlump.Thisusuallyproducesseverallumpsandalotofjuice.Othersaremoresystematic,peelingorangesverycarefullywithaknife,makingaspiralcutfromthetopoftheorangedowntothebottom.Ipersonallypreferamessonthetableandaquickorange,butthereyougo.

Left:Peelinganorangewithaknife.Middle:Thepeellaidoutflat.Right:Cornuspiral.

In2012LaurentBartholdiandAndréHenriqueswonderedwhatshapethepeelwouldformifitwerelaidoutflat.Usingathinknife,andbeingcarefultokeepthepeelthesamewidthallthewayalong,theyobtainedabeautifuldoublespiral.Itremindedthemofafamousmathematicaldoublespiral,variouslyknownasaCornuspiral,Eulerspiral,clothoid,orspiro.

Thiscurvehasbeenknownsince1744,whenEulerdiscoveredoneofitsbasicproperties.Itscurvature(1/rwhereristheradiusofthebest-fittingcircle)atanygivenpointisproportionaltothedistancealongthecurvetothatpoint,startingfromthemiddle.Thefurtheralongthecurveyougo,themoretightlyitcurves,whichiswhythespiralregionsbecomeevermorecloselywound.ThephysicistMarieAlfredCornucameacrossthesamecurveinthephysicsoflight,whenitdiffractsatastraightedge.Railwayengineershaveusedthecurvetoprovideasmoothtransitionbetweenastraightpieceoftrackandacircularone.

BartholdiandHenriquesprovedthattheresemblancebetweenorangepeelandthisshapeisnoaccident.Theywrotedownanequationdescribingtheorange-peelshapeforstripsofanyfixedwidth,andprovedthatwhenthewidthbecomesassmallasweplease,theshaperesemblestheCornuspiralevermoreclosely.Theyremarkedthatthisspiral“hashadmanydiscoveriesacrosshistory;oursoccurredoverbreakfast.”

Seepage253forfurtherinformation.

HowtoWintheLottery?Pleasenotethequestionmark.

TowinthejackpotintheUKNationalLottery(unimaginativelyrebrandedas‘Lotto’)youhavetochoosesixnumbersfromtherange1–49thatmatchthosedrawnonthedaybyaLottomachine.Thereareotherwaystowinsmallerprizes,butlet’ssticktothatone.Theballsaredrawninarandomorder,buttheresultsarethenconvertedtonumericalordertomakeiteasiertofindoutwhetheryou’vewon.Soadrawlike

13158484736

isreorderedas

81315364748

andinthiscasethesmallestnumberis8,thesecondsmallestis13,andsoon.Probabilitytheorytellsusthatwhenallnumbersareequallylikely,asthey

shouldbe,thenwithinachosensetofsix:

Themostlikelysmallestnumberis1.Themostlikelysecondsmallestnumberis10.Themostlikelythirdsmallestnumberis20.Themostlikelyfourthsmallestnumberis30.Themostlikelyfifthsmallestnumberis40.Themostlikelylargestnumberis49.

Thesestatementsarecorrect.Thefirstistruebecauseif1turnsup,thenitmustbethesmallest,nomatterwhatelsehappens.That’snotthecasefor2,however,becausethereisasmallchancethat1willturnupandsneakunderneathit.Thismakesitslightlylesslikelythat2willbethesmallestafterallsixballsaredrawn.

OK,thosearemathematicalfacts.Soitlooksasthoughyoucanimproveyourchanceofwinningifyoupick

11020304049becauseeachchoiceisthemostlikelynumbertooccurinthatposition.

Isthiscorrect?Seepage253fortheanswer.

TheGreenSocksCaperIncident“Youhavepassedthefirsttest,Doctor.Buttherealtestwillbetoobservehowyouhandleacriminalinvestigation.”

“Iamready,MrSoames.Whenshallwebegin?”“Notimelikethepresent.”“Iagree,wearebothmenofaction.Whichcaseshallitbe?”“Yourown.”“But—”“AmImistakeninthinkingthatalthoughyourreasonforcomingherewasto

seekemployment,youhavealsobeenthevictimofacrime?”“No,buthow—”“Whenyoufirstenteredthisroom,Iwasinstinctivelyawarethatyouwere

seekingmyassistance.Youwereattemptingtoconcealit,butIsawitinyourfaceandinyourbearing.WhenItestedmydeductionbyspeakingof‘thecrimethathasbeencommittedagainstyou’,yourreplywasevasive.Youstatedthatyouhadnotcomeprimarilyasaprospectiveclient.”

Watsupsighed,slumpinginhischair.“Iwasworriedthatmentioningmyowncasemighthaveanadverseeffectonyourdecisionaboutengagingmyservices,bysuggestingthatIwasmerelyseekingfreeadvice.Onceagainyouhaveseenthroughme,MrSoames.”

“Thatwasinevitable.Wemaydispensewithformalities.YoumaycallmeSoames.AndIshallcallyouWatsup.”

“Anhonour,Mr—er,Soames.”Watsup,clearlyupset,tookamomenttosteadyhimself.“Itisasimplematter,ofakindthatyouwillhaveencounteredmanytimesbefore.”

“Aburglary.”“Yes.Howdid—nomatter.Ithappenedearlierthisyear,andIimmediately

requestedprofessionalassistancefromyourneighbouracrosstheroad.Afteramonthinwhichhemadeabsolutelynoprogress,hedeclaredthatthematterwastootrivialtointeresthismightytalents,andshowedmethedoor.Hearingbyafortunateaccidentofyourownexploits,itoccurredtomethatyoumightsucceedwherethegreatluminaryhadfailed.”

ItwascleartoWatsupthathenowhadSoames’sfullattention.“Ivowtohelpyousolvethiscrime,toprovemyownworthtoyou,”Watsup

saidwithsomeemotion.“Ifwesucceed—nay,whenwesucceed—myhopesofamorepermanentengagementwillbeenhanced.Icanpayyounofee,butIcanoffermyownunpaidservicesfortwomonths.DuringwhichtimeIwillensureasteadyflowofclientsbysingingyourpraisestothegentry,enoughtokeepusbothfedandhousedinmoderatecomfort.”

“Iconfessthatsuchanarrangementdoeshavesomeappeal,”saidSoames.“Ihavebeenseekingwhatourtransatlanticfriendsrefertoasa‘sidekick’forsometime.Yourexposureofmylandlady’snosinessgivesmeadditionalconfidencethatyouwillfitthebilladmirably,butweshallsee.Er—speakingofbills,youdon’thappentohaveafive-poundnoteonyou,byanychance?MrsSoapsudsisalwayscomplainingabouttheunpaidrent…No,no,IseeyouareasstrappedforcashasI.Togetherweshallovercomeourmutualimpecuniosity.

“Now,tellmeofthecrime.”“AsIwassaying,itisasimplematter,”saidWatsup.“Myhousewas

burgled,andmypricelesscollectionofAl-Jebrianceremonialdaggers,representingthemajorityofmywealth,wasstolen.”

“Whenceyourpresentfinancialstate.”“Indeed.IhadplannedtohavethemauctionedatSotheby’s.”“Werethereanyclues?”“Justone.Agreensock,leftatthesceneofthecrime.”“Whatshadeofgreen?Whatmaterial?Cotton?Wool?”“Idonotknow,Soames.”“Thesethingsmatter,Watsup.Manyamanhasbeenhangedbecauseofthe

precisecolourofthedyeinasinglestrandofdarningwool.Orescapedthenooseforlackofsuchevidence.”

Watsupnodded,absorbingthelesson.“AlltheinformationIhavewasprovidedbythepolice.”

“Thatexplainsitspaucity,ofcourse.Prayproceed.”“Thepolicenarrowedresponsibilityforthecrimedowntothreemen:

GeorgeGreen,BillBrown,andWallyWhite.”Soamesnoddedthoughtfully.“The‘usualsuspects’,asIhadalready

surmised.TheyoperateintheBoswellStreetarea.”“HowdidyouknowIliveinBoswellStreet?”saidWatsupinastonishment.“Youraddressisonyourcard.”“Oh.Inanycase,oneofthosethreewasdefinitelythecriminal.Thepolice

madeenquiries,andfoundthateachmanhabituallyworeajacketandtrousers.”“Mostmendo,Watsup.Eventhelowerclasses.”“Yes.Butalso,socks.”Soamesprickeduphisears.“Afeatureperhapsofmildinterest.Itshowsthat

thesemenhaveanincomebeyondtheirmeans.”“I’msorry,Soames;Ireallydon’tsee—”“YouhaveneverencounteredMessrsBrown,Green,andWhite.”“Ah.”“Pleaseavoiddistractingremarks,Watsup,andgettothepoint.”“Apparentlyitwaseachman’sinvariablehabittodressingarmentswhose

colourswereexactlythesameonalloccasions.Subtletracesatthesceneofthecrime—”

“Yes,yes,”Soamesmutteredimpatiently.“Threadsadheringtothebrokenglass.Plainasthenoseonadonkey.”

“—uh,well,yes,asIwassaying,threads.Theseindicatedthatthethiefhadusedoneofhissockstomufflethesoundofbreakingwindow-glass,andthatthesockwasgreen.Witnessesconfirmedthatbetweenthemthethreemenworeonejacketofeachcolour,onepairoftrousersofeachcolour,andonematchingpairofsocksofeachcolour.Noneofthemworetwoormoregarmentsofthesamecolour—countingapairofsocksasasinglegarment,youappreciate,sinceevenruffianssuchasthesewouldnotwearsocksthatdonotmatch.Thatwouldbemostimproper.”

“Anddidyoudeduceanythingofconsequencefromthatinformation?”“Eachofthesuspectsmusthavewornexactlyonegarmentwiththesame

colourashisname,”saidWatsupinstantly.“Ifwededucethecolour,wefindthecriminal.”

Soamesleanedbackinhischair.“Verygood.Perhapswewillbeabletoworktogether.Anythingelse?”

“Icametotheconclusionthattheinformationtohandwasinsufficienttodeterminethecriminal.Thepoliceeventuallyadmittedasmuch,soIsuggestedtheyshouldseekfurtherevidence.”

“Anddidtheyfindany?”“AfterIhadofferedsomemorespecificadvice,theydid.”Watsuphanded

Soamesasheetofpaper.“Partofthepolicereport,”heexplained.Thedocumentread:

ExtractfromReportofInvestigationbyConstableJ.K.WugginsoftheHolbornDivisionoftheMetropolitanPoliceForce

1Brown’ssockswerethesamecolourasWhite’sjacket.2ThepersonwhosenamewasthecolourofWhite’strousersworesockswhosecolourwasthenameofthepersonwearingawhitejacket.3ThecolourofGreen’ssockswasdifferentfromthenameoftheperson

wearingthesamecolourtrousersasthejacketwornbythepersonwhosenamewasthecolourofBrown’ssocks.

“Andthereyouhaveit,”saidWatsup.“Ifwecandeterminethethief,thenthepolicewillbeabletoobtainasearchwarrant.Withlucktheywillfindmymissingdaggers,whichwouldamounttoincontrovertibleproofofguilt.Buttheyarestumped,andyouroverratedneighbourisasbaffledasI—whichiswhyhepretendsthatthecasehasnointerest.”

Soameschuckled.“Onthecontrary,mydearWatsup.Thankstoyourassiduousdevotiontopersuadingthepolicetoinvestigatethecircumstancesofthecrimeinsufficientdepth,thereisenoughinformationtodeterminetheguiltyparty.Thedeductionisofcourseelementary.”

“Howcanyoubesosure?”“Youwillcometoknowmymethods,”saidSoamesenigmatically.“Whoisthecriminal,then?”“Wewillfindoutwhenwemakethededuction.”Watsupproducedanew,fat,currentlyblank,notebook,andwrote:

MemoirsbyDrJohnWatsup(M.Chir.,R.M.C.S.,retd)

One:TheGreenSocksCaperSoames,readingthewordsupsidedown,saidquietly“Thisisnotapennydreadful,Watsup.”Watsupcrossedout‘Caper’andinserted‘Incident’.Then,pursinghislips,hebegantorecordtheirjointanalysis.Withafewhiccupsalongtheway,theidentityofthethiefsoonemerged.

Seepage254fortheanswer.“IshallsendInspectorRouladeatelegramimmediately,”Soamesdeclared.

“Hewillsendconstablestoraidtheman’spremises.Nodoubttheywillfindyourdaggersthere,sincethemanwehaveidentifiedisnotoriouslyslowtofencestolenproperty.Helikestogloat,Watsup,anerrorthathasputhimbehindbarsmorethanonce.

“Andthatwrapsupourfirstcasetogether!”Hisexcitementquicklysubsidedasheadded,“Yourassistancewasvital,butunfortunatelytheoutcomeofourdeliberationsdoesnotimproveourfinances,sinceitisyourcase.”

“Therewillbesomeimprovement.Iwillregainmydaggers.”“Ifearthepolicewillholdthemasevidenceuntilafterthetrial.Evenso,we

mayconsideritaharbingerofmoreprofitabletimestocome,eh,Watsup?”

ConsecutiveCubesThecubesofthethreeconsecutivenumbers1,2,3are1,8,27,whichaddupto36,aperfectsquare.Whatarethenextthreeconsecutivecubeswhosesumisasquare?

Seepage258fortheanswer.

AdonisAsteroidMousterian

ThreeofFarrell’smagicwordsquares

JeremiahFarrellpublishedsomeamazingmagicwordsquaresinTheJournalofRecreationalLinguistics33(May2000)83–92.Thesearesamples.Theentriesineachsquarearetwo-letterwordsappearinginstandarddictionaries.Thesamelettersappearineachrowandcolumn,andineachofthetwomaindiagonalsoftheorder4and5squares.Everyrowandcolumnisananagram(thoughnotameaningfulone)ofthesamedictionaryword,whichiswrittenunderneath.MousterianisastyleofflinttoolusedbysomeNeanderthals,bytheway.

Youmayfeelthatwordarrangementsarenotterriblymathematical.However,puzzlebuffstendtoenjoyboth,andIaminclinedtoseewordgamesascombinatorialproblemsposedwithirregularconstraints;namely,thedictionary.Butthesesquareshavemathematicalfeaturestoo.Ifnumbersaresuitablyassignedtoletters,andthenumberscorrespondingtoeachpairoflettersinagivensquareareaddedtogether,theresultingnumericalsquareisalsomagic.Thatis,thenumbersineveryrow,column,and(exceptforthe3×3square)diagonaladdtothesameamount.

Ofcourse,thispropertyholdsforanyassignmentofnumbers,exceptonthediagonalsofthe3×3square,becauseeachletteroccursexactlyonceineachrow,column,and(exceptforthe3×3square)diagonal.However,withthecorrectchoice,thenumbersrunfrom0–8,0–15,and0–24respectively.Theassignmentsaredifferentforeachmagicwordsquare.

Whichnumberscorrespondtowhichletter?Seepage258fortheanswer.

TwoSquareQuickies1Whatisthelargestperfectsquarethatuseseachdigit123456789exactlyonce?2Whatisthesmallestperfectsquarethatuseseachdigit123456789exactlyonce?

Seepage259fortheanswers.

CaughtClean-Handed

JohnNapier

JohnNapier,eighthLairdofMerchistoun(nowMerchiston,partofEdinburgh),isfamousforinventinglogarithmsin1614.Buttherewasadarkersidetohisnature:hedabbledinalchemyandnecromancy.Hewaswidelybelievedtobeamagician,andhis‘familiar’,ormagicalcompanion,wasablackcockerel.

Heusedittocatchservantswhowerestealing.Hewouldlockthesuspectinaroomwiththecockerelandtellthemtostrokeit,sayingthathismagicalbirdwouldunerringlydetecttheguilty.Allverymystical—butNapierknewexactlywhathewasdoing.Hecoatedthecockerelwithathinlayerofsoot.Aninnocentservantwouldstrokethebirdasinstructed,andgetsootontheirhands.Aguiltyone,fearingdetection,wouldavoidstrokingthebird.

Cleanhandsprovedyouwereguilty.

TheAdventureoftheCardboardBoxes*

FromtheMemoirsofDrWatsup

Withtherestorationofmyvaluableceremonialdaggers,andourpartnership’sgrowingreputationforcrackingtheuncrackable,solvingtheinsoluble,andunscrewingtheinscrutable,ourpersonalfinanceswereimprovingbytheday.TheeliteofEnglandwerevirtuallyqueuinguptoengageourservices,andmynotebookscontainmanyofmyfriend’ssuccesses:theMysteryoftheMissingMountain,theVaporisedViscount,andtheBald-HeadedLeague.Noneofthesecases,however,capturesSoames’stalentsintheirpurestform:hisabilitytodiscernsignificantaspectsofapparentlyordinaryobjectsandeventsthatfewotherswouldnotice.TheGiantBatofStAlbansnaturallyspringstomind,buttheramificationsofthecasearetooarcaneandcomplextodescribehere.

ThecuriouseventsofChristmas18—,however,suitmypurposeadmirably,anddeservewiderappreciation.(IamforcedtoconcealtheexactdateandmostofwhathappenedtoavoidembarrassmenttoafamousoperaticcontraltoandseveralCabinetMinisters.)

Iwasseatedatmywriting-desk,recordingdetailsofSoames’smostrecentcases,whilehecarriedoutaseeminglyendlessseriesofexperimentswithmyoldservicerevolverandvasesofchrysanthemums.OurseparateactivitieswereinterruptedbyMrsSoapsuds,whodepositedtwocardboardboxesofdifferentsizes,eachtiedwithribbons.“SomeChristmaspresentsforyou,MrSoames,”sheannounced.

Soameseyedthepackages.Theyborehisaddressandsomecancelledpostagestampswithillegiblepostmarks.Theywererectangularinshape…well,technicallyarectangleistwo-dimensional,sotheywereactuallyrectangularparallelepipeds.Cuboids.

Box-shaped.Hetookoutarulerandmeasuredtheirdimensions.“Remarkable,”he

muttered.“Andvery,verydisturbing.”Ihavelearnedtorespectsuchjudgements,howeverpeculiartheyfirstmay

seem.IstoppedthinkingofthepackagesasChristmaspresents,triedtodismissagrowingsuspicionthattheywerebombs,anddidmybesttoobserve.Finally,Irealisedthattheyhadbeentiedwithrathermoreribbonthanwasstrictly

necessary.

Left:Watsup’susualarrangementofribbons.Right:Thewayeachpackagewastied.

“Theribbonsformacrossoneveryfaceofthepackages,”Isaid.“WhenIwrapparcels,Inormallytietheribbonsothatitformsacrossonthetopandbottomfaces,andrunsverticallydowneachoftheotherfour.”

“Youdoindeed.”Clearlymyanalysiswasincomplete.Irackedmybrains.“Umm—thereisno

bow.”“Correct,Watsup.”Stillincomplete.Iscratchedmyhead.“ThatisallIcanobserve.”“Thatisallyoucansee,Watsup.Youhavenoticedeverythingexceptthe

crucialpattern.Ifearthatterribledeedsareafoot.”IconfessedthatIsawnothingterribleintwoChristmaspresents.Thena

thoughtstruckme.“Doyoumeanthattheboxescontainseveredbodyparts,Soames?”

Helaughed.“No,theyarealmostempty,”hesaid,pickingthemupandshakingthem.“ButsurelyyourealisethatthisparticulartypeofribboncanbeboughtonlyfromtheLadiesWilberforce?”

“Regrettably,no,butIbowtoyoursuperiorknowledge.Theestablishment,however,isfamiliartome.ItisahaberdasheryinEastcastleStreet.”Thepennydropped.“Soames!Thatiswherethatterriblemurderwascommitted!Itwas—”

“—inallthepapers.Yes,Watsup.”“Theevidencewasconvincing,butthebodyhasnotyetbeenfound.”Soamesnodded,hisfacegrim.“Itwillbe.”“When?”“ShortlyafterIopentheseboxes.”Heputonapairofglovesandsettoworkunwrappingthem.“Undoubtedly

thisistheworkoftheCartonari,Watsup.”WhenIstaredathimblankly,headded:“anItaliansecretsociety.Butitisbetterthatyouremainignorant.”Despiteallmyentreaties,herefusedtosaymore.

Heopenedbothboxes.“AsIsuspected.Oneisempty,buttheothercontainsthis.”Heheldupasmallrectangleofpaper.

“Whatisit?”Hepassedittome.“Aleft-luggageticket,”saidI.“Itmustbeamessage

fromthemurderer.Buttheserialnumberhasbeentornoff,andsohasthenameofthestation.”

“Onlytobeexpected,Watsup.He—forbythefootprintsinthebloodthecriminalwasassuredlyaman,andalargeoneatthat—istauntingus.Butweshallhavethebetterofhim.Thestationisofcourseobviousfromthearrangementoftheribbon.”

“Uh—pardon?”“Togetherwiththevalueofthestamps,whicheliminatesthepossibilityof

CharingCross.”Thismadelittlesense,soIpickedupapackageandcountedfiveone-shilling

stamps.“Anabsurdamounttopayforanemptypackage.”Isaid,puzzled.“Notifyouwishtosendamessage.Whatisanothernameforfive

shillings?”“Onecrown.”“Andacrownissymbolicof?”“Ourowndearqueen.”“Close,Watsup,butyouhavefailedtotakeaccountoftheshapeofthe

ribbon.”“Itisacross.”“Sothestampsindicate‘king’,notqueen.ThestationisKing’sCross,man!

Butthereismore.Answermethis,Watsup.Whydidthecriminalsendmetwolargeboxeswhenonewasempty?Onesmallenvelopewouldhavesufficedtosendaticket.”

AfteralongsilenceIshookmyhead.“Ihavenoidea.”“Theremustbesomethingsignificantabouttherelationshipbetweenthe

boxes.Andindeedthereis,asIrealisedassoonasImeasuredtheirsizes.”Hehandedmetheruler.“Usethis.”

Irepeatedhismeasurements.“Thelength,breadth,andheightofeachpackageisawholenumberofinches,”Isaid.“Otherwisenopatternspringstomind.”

Hesighed.“Youdidnotobservethestrangecoincidence?”“Whatstrangecoincidence?”“Bothpackageshavethesamevolume,andtheybothusethesametotal

lengthofribbon.Indeed,theirmeasurementsarethesmallestnonzerowholenumberswiththatproperty.”

“Whichleadsyoutoconclude—oh,ofcourse!Thevolumeandlengthtogethergivetheserialnumberontheticket.Therearetwodistinctwaysto

stringthemtogether,ofcourse,butwecaneasilycheckboth.”Soamesshookhishead.“No,no.Themurdererwouldhaveneededan

accompliceattheticketofficetoarrangethat,evenifaticketwiththatnumberexisted.Itismuchsimpler:hehasmarkedsomeitemofleftluggagewiththesetwonumbers.Inside,therewillbesomethingthattellsuswheretofindit.”

“Tofindwhat?”“Isitnotobvious?Thebody.”“Itakemyhatofftoyou,Soames,”Isaid.“Orwould,wereIwearingone.

Butwillfindingthebodyleadustothemurderer?”“Itwillbeusefulevidence,butinconclusive.However,thereismoretobe

gleaned.Sometimesacriminalbelieveshimselfsocleverthathedeliberatelyleavesclues,certainthattheinvestigatingauthoritieswillbetoostupidtonoticethem.TheCartonariareanarrogantbunch,andthisistypicalofthem.Now,thereisanaturalquestionthatleadsonfromtheremarkablearithmeticoftheseboxes.Whatisthesmallestsetofthreeboxeswithasimilarproperty?”

Histrainofthoughtbecameinstantlyapparenttome.“Youexpecttoreceivesuchboxesinthenearfuture!Withanothertornticket!Sothereistobeanothermurder,eh?”Istartedlookingformyrevolver.“Wemuststopit!”

“Ifearthatithasalreadybeencommitted,butwithgoodfortunewemayperhapspreventathirddeath.Tonightthemurdererwillbedepositingsomeitem—itmightbeanything—asleftluggageatoneofthemainLondonrailwaystations.Thenhewillsendustheboxes.Ifwecanworkoutthenumbersaheadoftime,wecanalertInspectorRoulade.Hewillsendpolicetoallofthemainlinestations.Theycannotcheckeverypassengerwhodepositsluggage,forthatwouldalertthecriminal,buttheycankeepawatchforanyonedepositinganitemwiththosethreenumbersmarkedonit,andarrestthem.Insidewillbethelocationofthesecondbody.Whenitisfound,theevidenceofguiltwillbeoverwhelming.”

Intheevent,itwasnotquitethatsimple,andSoamesandIhadtointerveneafterthepoliceletthemanslip.Fortunatelythethreepackages,whichdulyarrivednextdayintheafternoonpost,providednewclues,andwediscoveredthatthemurderwaspartofafarmoreextensiveplot.Thetortuouspathsdownwhichourdeductionsledus,andtheblood-curdlingsecretsthatweunearthed—Ispeakliterally—canneverbemadepublic,asIhaveexplained.Butwedideventuallycatchthecriminal.AndSoameshaspermittedmetorevealtheanswerstothetwoquestionsthatwerecentraltotheentireinvestigation.

Whatarethedimensionsofthetwoboxes?Whataretheyforthreeboxes?Seepage259fortheanswer.

Footnote

*ThisandallsubsequentcasesinvestigatedbySoamesandWatsuparereproduced(slightlyedited)fromTheMemoirsofDrWatsup:BeingaPersonalAccountoftheUnsungGeniusofanUnderratedPrivateDetective,Bromley,Thrackle&Sons,Manchester1897.

TheRATSSequence1,2,4,8,16,…Whatcomesnext?It’stemptingtoleaptoconclusionsandplumpfor32.ButsupposeItellyouthatthesequenceIhaveinmindactuallygoes

12481677145668

Nowwhatcomesnext?Thereisn’tauniqueanswer,ofcourse:byfindingsufficientlycontrivedrulesyoucanfitaformulatoanyfinitesequence.InMathematicsMadeDifficultCarlLinderholmhasanentirechapterexplainingwhyyoucanalwaysanswer‘whatcomesnextinthissequence’with‘19’.But,bearwithme:thereisasimpleruleforthesequenceabove.Thetitleofthissectionisaclue,butamuchtooobscureonetobeofanyusetoyou.

Seepage260fortheanswer.

BirthdaysAreGoodforYouStatisticsshowthatpeoplewhohavemostbirthdayslivethelongest.

LarryLorenzoni

MathematicalDatesInrecentyearsnumerousdateshavebeenassociatedwithaspectsofmathematics,basedonnumericalsimilarities,leadingtothatdatebeingdeclaredaspecialday.Nooneattachesanysignificancetosuchdays,asidefromthenumericalsimilarity.TheydonotforetelltheEndoftheWorldoranythingofthatkind—asfarasweknow.Nothingspecialhappensonthemasidefrommathematicalcelebrationsandoccasionalcommentsinthemedia.Butthey’refun—andanexcusetointerestthemediainmoresignificantmathematics.Oratleasttomentiontheword.

Amongthemarethefollowing.Manyareassociatedwithalternativedates,becauseintheAmericansystemfordates,themonthprecedestheday.IntheBritishsystem,thedaycomesfirst.Acertainamountofcalendriclicenceisallowed,suchasomittingzeros.

PiDay14March,Americansystemofdates:3/14.[π~3.14.]Quasi-officialdaysince1988inSanFrancisco.Recognisedbyanon-bindingresolutionintheUSHouseofRepresentatives.

PiMinute1.59onMarch14(Americansystem).3/141:59.[π~3.14159.]

Moreaccuratelystill:1.59and26seconds.3/141:59:26.[π~3.1415926.]

PiApproximationDay22July,Britishsystemofdates:22/7.[π~22/7.]

123456789DaySorry,youmissedit.Thisonce-in-a-lifetimeeventoccurredon7August2009(Britishsystem),or8July2009(Americansystem),shortlyafter12.34p.m.Thedateandtimewere:12:34:567/8/(0)9.Butsomeofyoumightwitness1234567890Dayin2090.

OnesdayYoumissedthistoo.Itoccurredon11November2011(eithersystem)at11minutes,11secondspast11o’clock.Thedateandtimewere:11:11:1111/11/11.

TwosdayAsIwrite,thiswillhappenafewyearsfromnow.Thisisyouropportunity!2February2022:22:22:222/2/22.

PalindromeDayApalindromereadsthesameinreverse—asin‘evilratsonnostarlive’.20February2002at8.02p.m.(Britishsystem,24-hourclock):20:0220/02/2002.

Thisrepeatsthesamepalindromethreetimes.Whenwillthenextsuchdaybe,ontheBritishsystem?Whatwasthenextdateaftertheoneabove,againusingtheBritishsystem,onwhichthewholethingwaspalindromic?

FortheanswersSeepage261.

FibonacciDay(Shortversion)3May2008(Britishsystem),5March2008(Americansystem):3/5/(0)8.

FibonacciDay(Longversion)5August2013(Britishsystem),2minutes3secondspast1o’clockon8May2013Americansystem):1:2:35/8/13.

PrimesDay2March2011(Britishsystem),3February2011(Americansystem):2:35/7/11.

TheHoundoftheBasketballsFromtheMemoirsofDrWatsup

“Aladytoseeyou,MrSoames,”saidMrsSoapsuds.SoamesandIsprangtoourfeet.Awomanofindeterminateageentered—

indeterminatebecausesheworeadarkveil.“Thereisnoneedtodisguiseyourself,LadyHyacinth,”saidSoames.Withagasp,shepulledoffherveil.“How—”“TheextraordinaryeventsatBasketHallhavebeenintheheadlinesfora

week,”saidSoames.“Ihavebeenfollowingthecaseclosely,andmycompetitoracrosstheroadhasmadenoprogress.Itwasonlyamatteroftimebeforeyousoughtmyservices.Also,Irecognisedyourdriver’shat,whichisquiteuniqueamongtheservantsofthearistocracy.”

“Basquet,notBasket,”saidLadyHyacinthwithasniff,makingthewordsoundFrench.

“Comenow,madam,”saidSoames.“ThehousehasbeenintheBasketfamilyforsevengenerations,eversinceHonoriaThumpingham-MaddelymarriedtheThirdEarl.”

“Well,yes,butthatwasthen.Thespellingandpronunciationhave…er—”“Beenmodernised,”Iinterjected,hopingtosmoothincreasinglytroubled

watersofmutualdisaffection.AtthesametimeIgaveSoamesasharplook,unobservedbyHerLadyship.Tohiscredit,hetookmypoint.

“Itwasagiganticblackhound!”shesuddenlycried,thewordssoundingasiftheywerebeingtornfromherthroat.“Withgreatslaveringjawsthatdrippedblood!”

“Yousawit?”“Well,no,buttheboywholooksafterthepiglets…Nicky,that’sthename.

OrisitRicky?Anyway,hesaidhecaughtaglimpseofthevilehorrorjustasitdisappeared.”

“Inthedark,”Soamespointedout.“Fromadistanceofonehundredandseventyyards.MichaelJenkinsisshort-sighted.Butnomatter,theevidencewillleadustothetrutheventually.Igatherthattheanimaldidnoharmtoanyhumanbeing?”

“Well,no,”sheagreed.“Notdirectly,thoughmypoorhusband…Yousee,

thehoundruinedatraditionthatgoesbackeventobeforetheThirdEarlofBask—Basquet.”

Ibelatedlyrememberedmyetiquette.“DrJohnWatsup,atyourservice,madam.IregretthatunlikemycompanionIhavenotbeenfollowingthenews.Ifyouwouldbesokindastoenlightenme?”

“Ah.Yes.Um.”Shegatheredherskirtsandherwits.“ItwasafewnightsbeforeMidwinter’sEve,andmyhusbandEdmund—that’sLordBasquet,ofcourse—hadarrangedtwelveantiquestonespheres—”

“Knownforcenturiesasbasketballs,”Soamesinterrupted.“Well,yes,butwecan’tmoderniseeverything,MrSoames.Thereare

traditions.Anyway,myhusbandhadarrangedtheballsonthelawnofourstatelyhomeinanage-oldfamilysymbol.Onlytheheirtothemalelineknowsexactlywhatthesymbolis,andnooneelseispermittedtoobservetheceremony,butovertheyearsithasbecomecommonknowledgethatitconsistsofsevenstraightrowsofballs,withfourineachrow.

“EdmundwasrehearsingforaceremonywhichmustbecarriedoutwithoutfaileveryMidwinter’sEve.Butwhenweawokethefollowingmorning,wewerehorrifiedtofindthatsomeoftheballshadbeenmoved!”

“ButyouhavesaidthatnoonesaveLordBasquetispermittedtoseethearrangement,”Soamesobjected.

“Thecircumstanceswereexceptional.HisLordshipwenttorecovertheballs,butfailedtoreturn.Eventuallyoneoftheserving-maidswassenttofindhim—Laviniaisblind,yousee,butverycapable.ShecamebackintearsscreamingthathisLordshipwaslyingontheground,notmoving.Fearinghewasdead,therestofusdisobeyedthetime-honouredinjunctionandrushedtothescene.IwasjustintimetohearEdmundgasp‘Moved!’Thenhewasstill.Hehasbeeninastuporeversince,MrSoames.Itismostdistressing.”

“Moved,”saidI.“Inwhatmanner,madam?”“Nolongerinthesameplace,DrWatsup.”“Imean,movedwhere?”“Theynowformedastar,DrWatsup.”“Yes!Onehavingonlysixstraightrowswithfourballsineachrow,”said

Soames,sketchingrapidlyonasheetofpaper.“Thisfacthasbeenwidelyreportedandhastheringoftruth,foritisunlikelytohavebeeninvented,beingtoomentallytaxingfortheyellowpress.ItalsoprovesthatwecouldhavededucedthattheballshadbeenmovedwithouthavingtorelyonhisLordship’slastgasp—”

Theballsafterthehoundhadmovedthem

“Lastgasptodate,”Isaidhurriedly,beforeSoamestriggeredafreshwaveofwails.

“Couldyounotmovethemback?”Ienquired,whenHerLadyshiphadregainedsomeofhercomposure.

“Nay!”shecried.IhavelongremarkedthattheEnglisharistocracyhaveanoticeabletendencytoresemblehorses.

“Whynot?”“AsItoldyou,onlyhisLordshipknowstheexactarrangementthattradition

requires,andthedoctorssayhemayneverrecover!”“Weretherenotmarkswheretheballshadoriginallyrested?”“Perhaps,buttheywereobscuredbythetracksoftheterriblehound!”“Iwillbringmymostpowerfulmagnifyingglass,then,”saidSoames,witha

straightface.Athoughtmusthavestruckhim,becausehesuddenlyfroze.“Yousaid‘must’.”

“Idid?When?”“Afewminutesagoyousaidthattheceremonymustbecarriedoutwithout

faileveryyear.Ithasjustoccurredtomethatyourchoiceofwordsmaybesignificant.Explainthem.”

“AccordingtoanancientTransylvanianprophecy,ifthetwelvebasketballsarenotdeployedcorrectlyonMidwinter’sEve,theHouseofBask—er,Basquet—willfall,andbeutterlydestroyed!Andwehaveonlythreedaystodoit!Oh,woe!”Shebegantosob.

“Calmyourself,madam,”saidI,waftinganopenbottleofsmelling-saltsbeneathhernose.“PleaseacceptmycondolencesforHisLordship’sunfortunatecondition,andmyassurancesasamedicalmanthatthereissomefaintchance,beiteversoslight,thatitmayundergosomemiraculousimprovementinduecourse.”Ihavelongpridedmyselfonmyimpeccablebedsidemanner,which

putsthatofmyfriendSoamestoshame,butonthisoccasion,unaccountably,HerLadyship’sresponsewastoredoublehersobbing.

Soamespacedtheroom,hisfacedrawn.“Isitonlytheformofthearrangementthatmatters,YourLadyship?Ordoestheorientationmakeasignificantdifference?”

“Ibegyourpardon?”saidshe,shakingherheadasiftoclearhermind.“Ifthearrangementwerecorrectexceptforarotation,withoutchangingthe

relativepositionsoftheballs,wouldthatbeexpectedtotriggerthedireeventspredicted?”Soamesclarified.

LadyBasquetpaused,consideringthematter.“No.Definitelynot.IrecallWillyWillikins—that’stheheadgardener—suggestingtomyhusbandthatfromtimetotimehemightcaretopointthearrangementinadifferentdirection,toavoiddamagingthelawn.Edmundraisednoobjection.”

“Thatisexcellentnews!”saidSoames.“Yes.Excellent,”Iechoed,havingnotthefaintestideawhymydetective

friendwassopleased.Or,forthatmatter,whathisquestionmeant.“Werethereanysignsofhumanintervention?”Soamesasked.“No.TheheadgardenersworethatnohumanbeingsaveEdmundhadset

footonthelawn.YoungDicky—”“Micky.”“Vickysawtheterriblehound,butevenheobtainedonlyafleetingglimpse

asitleapedovertheperimeterofthewalledgarden.Wehavesomedelightfulpeonies,MrSoames,althoughtheyarenotinbloomatthistimeof—”

“Iwilltakethecase,”saidSoames.“IfYourLadyshipcarestoreturntoBasquetHall,IandmycolleaguewillarriveonThursdaybytheearliestpossibleslowtrain.”

“Nosooner,MrSoames?ButThursdayisMidwinter’sEve!Theballsmustbeplacedcorrectlybeforethesunsets!”

“IregretthatIamdetaineduntilthenbyasmallmatterinvolvingthreeEasternpotentates,sixhundredthousandarmedwarriors,twodisputedborders,andastolencasketofemeraldsandsapphiresbelongingtoanobscureandarcaneancientreligiousorder.Andaflattenedcopperthimble,whichIbelieveholdsthekeytotheentireaffair.However,IassureyouthatIamconfidentthatyourcasewillbesettledsatisfactorilybeforesunsetonThursday.”

Herprotestationsnotwithstanding,Soamesremainedadamant,andeventuallyLadyHyacinthBasquetdeparted,snufflingquietlyintoalacekerchief.

Aftershehadleft,IaskedtowhichmatterSoameshadbeenreferring,forIhadheardnothingofthiscase.“Asmallfabricationonmypart,Watsup,”he

confessed.“Ihaveticketsfortheoperathisevening.”Wearrivedmid-afternoonontheThursday,tobemetatthestationbya

groomdrivingagovernesscart.Orpossiblyagovernessinagroomcart,mynotesbeingillegibleonthispoint.LordBasquetremainedinacoma,wewereinformed.WithinabarehalfhourwehadarrivedattheHall,andSoameswasinspectingtheextensivelawnsusinganunusuallylargemagnifyingglass,ahairbrush,andaprotractor.

“Achanceforyoutoexerciseyourdeductiveskills,Watsup,”hesaid.“Iseesomedisturbanceofthegrass,Soames.”“CorrectWatsup.Thetracksarehighlycomplex,buttheyaremainly

superimposedpaw-printsof—”heloweredhisvoicesothatIalonecouldhear—“aminiaturepoodle.”Inhisnormalvoicehecontinued:“Iamunabletodiscerntheoriginalplacingoftheballs,butunlessIamverymuchmistaken—andIneveram—itisclearthattheanimalmovedpreciselyfouroftheballs.”

“Isthatsignificant,MrSoames?”LadyBasquetaskednervously,cradlingaminiaturepoodleinherarms.

Soamesglancedmyway.“Itis…possible…”Ibegan,andsawSoamesnodimperceptibly.Well,not

totallyimperceptibly,youappreciate,sinceifithadbeenimperceptibleIwouldnothaveseenit.Takingthisasveiledencouragement,Ihazardedaguess:”…thatthisconditionwillmakeitpossibletodeducetheoriginallayout.”

“Anddoesit?”sheenquired,ahopefullookonherface.

Whatwastheoriginalarrangementofthebasketballs?Seepage261fortheanswer.

DigitalCubesThisisanoldie,butitopensupalessfamiliarquestion.Thenumber153isequaltothesumofthecubesofitsdigits:

13+53+33=1+125+27=153

Therearethreeother3-digitnumberswiththesameproperty,excludingnumberslike001withaleadingzero.Canyoufindthem?

FortheanswerSeepage262.

NarcissisticNumbersThecubespuzzlehassomenotoriety,becausein1940thefamouspuremathematicianGodfreyHaroldHardywrote,inAMathematician’sApology,thatsuchpuzzleshavenomathematicalmeritbecausetheydependonthenotationemployed(decimal)andarelittlemorethancoincidences.However,youcanlearnquitealotofusefulmathematicsbytryingtosolvethem,andgeneralisations(forexample,tonumberbasesotherthan10)avoidthenotationalissue.

Oneoftheoffshootsofthispuzzleistheconceptofanarcissisticnumber,whichisdefinedtobeanumberthatisequaltothesumofthenthpowersofitsdecimaldigitsforsomen.Thetermn-narcissisticnumberisusedifwewanttomakenexplicit.

FourthPowersofDigits(4-NarcissisticNumbers)Write[abcd]forthenumberwithdigitsa,b,c,dtodistinguishthisfromtheproductabcd.Thatis,[abcd]=1000a+100b+10c+d.Wemustsolve

[abcd]=a4+b4+c4+d4

whereallunknownsliebetween0and9.Thisisbynomeansatrivialtask.Tryit!

Seepage263fortheanswer.

FifthPowersofDigits(5-NarcissisticNumbers)Thistimetheproblemistosolve

[abcde]=a5+b5+c5+d5+e5

andasyou’dexpect,that’sevenharder.

Seepage263fortheanswer.

HigherPowersofDigits(n-NarcissisticNumbers,n≥6)Itiseasytoprovethatn-narcissisticnumbersexistonlyforn≤60,becausewhenevern>60wehaven:9n<10n–1.In1985DikWinterprovedthatthereareexactly88narcissisticnumberswithnonzerofirstdigit.Forn=1theyarethetendigits(weinclude0becauseitistheonlydigitinthiscase).Forn=2they

don’texist.Forn=3,4,5seetheanswerstoDigitalCubes(page262)andtheabovetwoproblems.Forn≥6theyare:

nn-narcissisticnumbers

6 5488347 1741725,4210818,9800817,99263158 24678050,24678051,885934779 146511208,472335975,534494836,91298515310 467930777411 32164049650,32164049651,40028394225,42678290603,44708635679,

49388550606,82693916578,9420459191414 2811644033596716 4338281769391370,433828176939137117 21897142587612075,35641594208964132,3587569906225003519 1517841543307505039,3289582984443187032,4498128791164624869,

492927388592808882620 6310542598859969391621 128468643043731391252,44917739914603869730723 21887696841122916288858,27879694893054074471405,

27907865009977052567814,28361281321319229463398,35452590104031691935943

24 174088005938065293023722,188451485447897896036875,239313664430041569350093

25 1550475334214501539088894,1553242162893771850669378,3706907995955475988644380,3706907995955475988644381,4422095118095899619457938

27 121204998563613372405438066,121270696006801314328439376,128851796696487777842012787,174650464499531377631639254,177265453171792792366489765

29 14607640612971980372614873089,19008174136254279995012734740,19008174136254279995012734741,23866716435523975980390369295

31 1145037275765491025924292050346,1927890457142960697580636236639,2309092682616190307509695338915

32 17333509997782249308725103962772

33 186709961001538790100634132976990,186709961001538790100634132976991

34 112276328532937254159282290020459335 12639369517103790328947807201478392,

1267993778027227856630388559419692237 121916721962543412156973580360996601938 1281579207836605995509977054529612936739 115132219018763992565095597973971522400,

115132219018763992565095597973971522401

Piphilology,Piems,andPilishNow,IwishIcouldrecollectpi.

‘Eureka!’criedthegreatinventor.Christmaspudding;ChristmaspieIstheproblem’sverycentre.

See,Ihavearhymeassistingmyfeeblebrain,itstaskssometimesresisting.

HowIwishIcouldenumeratepieasily,sinceallthesehorriblemnemonicspreventrecallinganyofpi’ssequencemoresimply.

Thelastonegivesthegameaway:thesearemnemonics—memoryaids—forπ.There’sevenawordforthis:piphilology.Countthelettersinsuccessivewords:3,1,4,1,5,…

SomeofthemanymnemonicsforπwerediscussedinCabinetofMathematicalCuriosities;herewerecallone(theFrenchmnemonicbelow)andlookatafewmore.Therearehundreds,inmanylanguages:see

http://en.wikipedia.org/wiki/Piphilologyhttp://uzweb.uz.ac.zw/science/maths/zimaths/pimnem.htm

OneofthemostfamousistheFrenchalexandrine(apoeticmetre),whichbegins:

Quej’aimeàfaireapprendreUnnombreutileauxsages!GlorieuxArchimède,artisteingenieux,Toi,dequiSyracuseloueencorelemérite!

andgoesonuntilitgetsto126places.IespeciallyrecommendthePortuguese:Souomedoetemorconstantedomeninovadio.

(Iamtheconstantfearandterroroflazyboys.)TheRomanian:

Asaebineascrierenumitulsiutilulnumar.(That’sthewaytowritethefamousandusefulnumber.)

hasthemeritofdirectnessandsimplicity.Thepoemsareknownaspiems.The32nddecimalplaceofπis0,anda

wordoflength0isinvisible.However,therearewaysroundthisobstacle.InPilish,thecryptographicsystemnormallyusedforπmnemonics,aten-letterwordcountsas0.MikeKeith’s‘ASelf-ReferentialStory’,whichencodes402

digitsofπ[TheMathematicalIntelligencer8(No.4)(1986)56–57]employedadifferentsetofrules.ThelongestexamplesIknowof(todate,GuinnessBookofWorldRecordsherewecome,ifIknowmyreaders)aretheshortstory‘Cadaeiccadenza’(3,834digits)andthebookNotAWake(10,000digits),alsobyKeith.Thebookbegins:

NowIfall,atiredsuburbianinliquidunderthetreesDriftingalongsideforestssimmeringredinthetwilightoverEurope.Soscreamwiththeoldmischief,askmeanotherconundrumAboutbitternessofpossiblefortunesnearalandscapeItalian.Alittlehappinessmaysometimesintervenebutusuallyfades.Amissionarycries,strivingtounderstandworthless,tediouslife.Monotony’slostamidoceanmovementsAsthebewilderedsailorshesitate.Ibecomesalt,Submergingpeopleindazzlingoceansofenshroudedunbelief.Christmasornamentsconspire.Beautyis,somewhatinevitablynow,bothFeelingsoffaithandeyesofrationalism.

Hereten-letterwordscountas0,andlongeronescountastwodigits;forexample,a13-letterwordcountsas13.

ForawealthofrelatedinformationandotherexamplesseeKeith’swebsitehttp://cadaeic.net

Clueless!FromtheMemoirsofDrWatsup

AsIflickthroughthewell-thumbedpagesofmynotebooks,mymemoriesaredrawntoinnumerablemysteriesthatSoamessolvedbyobservingcluessosubtleastoeludelesserminds,suchastheAdventureoftheSussexUmpire(aremarkablelocker-roommysterywhosepivotalfeaturewasaprematurelyscuffedcricketball),theCowwiththeCrumpledHorn,theAttemptedTripleMurderoftheDiminutiveSwine,andtheAffairoftheMissingTart.Butamongthemonestandsout:amysterywhosesolecluewastheabsenceofanyclueswhatsoever.

Itwasadamp,gloomyTuesday,andthestreetsofcentralLondonwerethickwithsmokeandfog.Wehadabandonedactivepursuitoftheperpetratorsofcrimeforaperiodofintrospectionbeforeawarmfire,accompaniedbyampleglassesofanamusinglypresumptuousclaret.

“Isay,Soames,”Iremarked.Mycolleaguewassearchingthroughathickstackofphotographicplatesof

hoofprintsinmud,producedusingEastman’snewimprovementofMaddox’sgelatineprocess.Hisresponsewasanirritated“Haveyouseenmycollectionofcarthorsephotographsanywhere,Watsup?”,butIcontinueddoggedly.

“Thispuzzlehasn’tgotaclue,Soames.”“It’snottheonlyone,”hemuttereddarkly.“No,Imean—ithasn’tgotanyclues.”IcouldseeInowhadhisattention.Hetookthenewspaperfrommy

outstretchedhand,andglancedatthediagram.

Cluelesspuzzle

“Theunstatedrulesareobvious,Watsup.”“Why?”“Theymustbesimpleenoughtomotivatethewould-besolver,yetleadtoa

sufficientlychallengingproblemtoretaintheirinterest.”“Nodoubt.Sowhataretherules,Soames?”“Clearlyeachrowandeachcolumnmustcontaineachnumber1,2,3,4once

only.”“Ah.Itisacombinatorialpuzzle,atypeofLatinsquare.”“Yes,butthereismore.Thetworegionsoutlinedinthickblacklinesare

obviouslyimportant.Iconjecturethatthenumbersineachregionmusthavethesametotal…Yes,thatleadstoauniquesolution.”

“Oh.Iwonderwhattheansweris.”“Youknowmymethods,Watsup.Usethem.”Andhereturnedtohis

photographicplates.

Seepage263fortheanswer.Formorecluelesspuzzles,Seepage74.

ABriefHistoryofSudoku

ModernreaderswillrecogniseWatsup’spuzzleasavariantofsudoku.(Ifyou’vejustcomebackfromaforty-yeartriptoProximaCentauri,thisisa9×9arraydividedintonine3×3blocks,withsomenumbersinserted.Youhavetofillintherestsothateveryrow,everycolumn,andeveryblockcontainseachdigit1–9.)

LoShu,centre,onthebackofasmallturtle,surroundedbytheChinesezodiacandthedivinatorytrigramsoftheIChing,allcarriedbyalargerturtle,whichaccordingtomythfirstrevealedthetrigrams

Similarbutsignificantlydifferentpuzzleshavealonghistory,whichgoesrightbacktotheChineseLoShu,amagicsquareallegedlyseenonthebackofaturtlearound2100BC.JacquesOzanam’s1725RécréationsMathématiquesetPhysiquesincludedapuzzleaboutplayingcardsthatgetsslightlyclosertosudoku.Takethe16courtcards(ace,king,queen,jack)andarrangetheminasquaresothateachrowandcolumncontainsallfourfacevaluesandallfoursuits.KathleenOllerenshawshowedthatthereare1152solutions,whichreducetojusttwobasicallydifferentonesifweconsidertwosolutionstobethesameifonecanbeobtainedfromtheotherbypermutingthefacevaluesandsuits.Thereare24×24=576waystodothistoanygivensolution,and1152/576=2.

Canyoufindthesetwobasicallydifferentsolutions?Seepage265fortheanswer.

In1782Eulerwroteaboutthethirty-sixofficersproblem:cansixregiments,eachhavingsixofficersofdifferentranks,bearrangedina6×6squaresothat

eachrowandcolumncontainseveryrankandeveryregiment?SucharrangementswerecalledGraeco-LatinsquaresbecauseLatinlettersA,B,C,…andGreeklettersα,β,γ,…couldbeusedtodenotetheranksandregiments.HefoundmethodsforconstructingGraeco-Latinsquareswhoseorder(thesizeofthesquare)isoddordoublyeven:amultipleoffour.

Anorder5Graeco-Latinsquare

Eulerconjecturedthatnosuchsquaresexistwhentheorderisdoublyodd:twiceanoddnumber.Thisisobviousfororder2,andGastonTarryproveditin1901fororder6.However,in1959RajChandraBoseandSharadchandraShankarShrikhandeusedacomputertofindaGraeco-Latinsquareoforder22,andErnestParkerfoundoneoforder10.ThethreeofthemthenprovedthatEuler’sconjectureisfalseforalldoublyoddordersgreaterthanorequalto10.

Squaren×narrayssuchthateachrowandcolumncontainsallofthenumbers1—n(eachnecessarilyappearingonce)becameknownasLatinsquares,andGraeco-LatinsquareswererenamedorthogonalLatinsquares.Thesetopicsarepartofthebranchofmathematicsknownascombinatorics,andtheyhaveapplicationstoexperimentaldesign,theschedulingofcompetitions,andcommunications.

AcompletedsudokugridisaLatinsquare,butthereareextraconditionsonthe3×3blocks.In1892theFrenchnewspaperLeSièclepublishedapuzzleinwhichsomeofthenumberswereremovedfromamagicsquareandreadershadtoinsertthecorrectones.LaFrancecameveryclosetoinventingsudokubyusingmagicsquarescontainingonlythedigits1–9.Inthesolutions,3×3blocksalsocontainedtheninedigits,butthiswasnotmadeexplicit.

SudokuinitsmodernformwasprobablyintroducedbyHowardGarnsandpublishedanonymouslyin1979byDellMagazinesunderthename‘numberplace’.In1986theJapanesecompanyNikolipublishedpuzzlesofthiskindinJapanunderthenotterriblyeye-catchingnamesūjiwadokushinnikagiru(‘thedigitsarelimitedtooneoccurrence’).Thenamewasthenshortenedtosūdoku.

TheTimesbeganpublishingsudokupuzzlesintheUKin2004,afterbeingcontactedbyWayneGould,whohaddevisedacomputerprogramtoproducesolutionsrapidly.In2005itbecameaworldwidecraze.

Hexakosioihexekontahexaphobia

Thisisthefearofthenumber666.In1989,whenPresidentRonaldReaganandhiswifeNancymovedhouse,

theychangedtheaddressfrom666St.CloudRoadto668St.CloudRoad.Thismaynothavebeenagenuinecaseofhexakosioihexekontahexaphobia,however,becausetheymightnothavefearedthenumberassuch—theywerejusttakingstepstoavoidobviousaccusationsandpotentialembarrassment.

Ontheotherhand…WhenDonaldRegan,ChiefofStafftoPresidentReagan,publishedhismemoirsinthe1988FortheRecord:FromWallStreettoWashington,hewrotethatNancyReaganhadregularlytakenadvicefromastrologersJeaneDixonand,later,JoanQuigley.“VirtuallyeverymajormoveanddecisiontheReagansmadeduringmytimeasWhiteHouseChiefofStaffwasclearedinadvancewithawomaninSanFranciscowhodrewuphoroscopestomakecertainthattheplanetswereinafavorablealignment.”

Thenumber666hasoccultsignificancebecauseitisstatedtobetheNumberoftheBeastintheBookofRevelation13:17–18(KingJamestranslationoftheBible):“Andthatnomanmightbuyorsell,savehethathadthemark,orthenameofthebeast,orthenumberofhisname.Hereiswisdom.Lethimthathathunderstandingcountthenumberofthebeast:foritisthenumberofaman;andhisnumberisSixhundredthreescoreandsix.”

ItisgenerallyassumedthatthereferenceistothenumerologicalsystemknownasgematriainHebrewandisopsephyinGreek,inwhichlettersofthealphabetareassociatedwithnumbers.Severalsystemsarepossible:numberthelettersofthealphabetconsecutively,ornumberthem1–9,10–90,and100-900orwherevertheprocessstops(whichistheancientGreeknumbernotation).Thenthesumofthenumbersassociatedtothelettersinaperson’snameisthenumericalvalueofthatname.

InnumerableattemptshavebeenmadetodeducewhotheBeastwas.TheyincludetheAntichrist(writteninLatinintheaccusativecaseasAntichristum),theRomanCatholicChurch(identifiedwithoneofthePope’stitles:VicariusFiliiDei),andEllenGouldWhite,founderoftheSeventhDayAdventists.Howcome?Well,countingonlytheLatinnumeralsinhername,youget

addingto666.IfyouthinktheBeastwasAdolfHitler,youcan‘prove’itby

startingthenumberingatA=100:

Basically,youchooseyourhate-figure,basedonyourownpoliticalorreligiousviews.Thenyoutwistthenumbering,andifnecessarythename,tofit.

However,allofthesedeductionsmaybebasedonamisapprehension—otherthanthebeliefthatanyofitactuallymatters—becauseithasnowbecomeapparentthat666maybeanerror.Around200ADthepriestIrenaeusknewthatseveralearlymanuscriptsstatedadifferentnumber,butheattributedthistomistakesbythescribes,assertingthat666wasfoundin“allthemostapprovedandancientcopies”.Butin2005scholarsatOxfordUniversityusedcomputerimagingtechniquestoreadpreviouslyillegibleportionsoftheearliestknownversionofRevelation,number115ofthepapyrifoundattheancientsiteofOxyrhynchus.Thisdocument,whichdatestoaboutAD300,isthoughttobethemostdefinitiveversionofthetext.ItgivestheNumberoftheBeastas616.

Once,Twice,ThriceThesquarearray

192384576

useseachoftheninedigits1–9.Thesecondrow384istwicethefirstrow192,andthethirdrow576isthreetimesthefirstrow.

Therearethreeotherwaystodothis.Canyoufindthem?

Seepage265fortheanswer.

ConservationofLuck“AfriendofminewonsevenmillionontheLotto,”saidthechapnexttomeinthegym.“That’stheendofmychances.Youcan’twinifyouknowsomeonewhohas.”

ThereareasmanyurbanmythsabouttheUKNationalLotteryastherearelegsonamillipede,butI’dnotcomeacrossthisonebefore.Itsetmewondering:whydopeoplesoreadilybelievethiskindofthing?

Thinkaboutit.Inorderformyfriend’sbelieftobetrue,theLottomachinehastosomehowbeinfluencedbyhisnetworkoffriendsandacquaintances.Ithastoknowwhetheranyofthemhaswonbefore,andthentakestepstoavoidhisparticularchoiceofnumbers,whichmeansthatitalsohastoknowwhathehaschosen.Infact,allelevenLottomachinesusedintheUKlotterymustknowthis,becausetheoneusedeachweekisitselfchosenatrandom.

SinceaLottomachineisaninanimatemechanicaldevice,thisdoesn’tmakeagreatdealofsense.

Eachweek,thechanceofanyparticularsetofsixnumberswinningthejackpotis1in13,983,816.That’sbecausetherearethatmanypossiblecombinationsofnumbers,andeachisequallylikelytooccur.Ifnot,themachinewouldbebiased,anditisdesignedtoavoidthat.Yourchancesofwinningdependonlyonwhatyou’vechosenthisweek,notonwhatsomeoneyouknowoncedid.Thelikelyamountyouwillwindoesdependonotherpeople,however:ifyouhitthejackpotandotherschosethesamenumber,youallhavetosharetheprize.Butthat’snotwhatmyfriendwasworriedabout.

Thereasonswhysomeofusbelievethiskindofmythlieinhumanpsychologyratherthanprobabilitytheory.Onepossiblereasonisanunconsciousbeliefinmagic,heremanifestingitselfasluck.Ifyouthinkthatluckisarealthingthatpeoplepossess,anditimprovestheirchances,andifyouthinkthatthereisonlyacertainamountoflucktogoround,thenperhapsyourfortunatefriendhasusedupalltheluckinyourneighbourhood.Whichinthisinstanceseemstobeyoursocialnetwork.OMG!Canyoutweetyourluckaway?PutyourluckonFacebookforyourso-calledfriendstosteal?It’sanightmare!

Ormaybetheunderlyingideaislikethepersonwhotakesabombonboardanaircraftwhenevertheytravel,onthegroundsthatthechanceoftwobombsonthesameplaneisinfinitesimal.(Thefallacyisthatyouchosetobringitonboard.Thishasnoeffectonthechanceofsomeoneelsehavingdonethesame,unknowntoyou.)

It’struethatmostLotterywinnersdonothavefriendswhoarealsowinners.Soit’seasytodeducethatifyouwanttobeawinner,youshouldavoidhavingsuchfriends.Actually,mostLotterywinnerslackwinningfriendsforthesamereasonthatmostloserslackwinningfriends:thereareveryfewwinnersbutavastnumberoflosers.

Agreed,youhavetobeinittowinit.Anacquaintanceofminewonhalfamillionpounds,andwouldnothavebeenpleasedifI’dadvisedhernottobother,andthenherusualnumberscameup.

Knowingthattheoddsarefirmlyagainstme,andnotfindingtheallegedthrillofgamblingworththevirtualcertaintyofthrowingmyhard-earnedmoneydownthedrain,IneverbetontheLottery.ButIhave,overtheyears,beeninadvertentlybettingonalotteryofmyown:writingabestseller.Ihaven’twonthejackpot,butI’vedefinitelycomeoutahead.AfewyearsagotheauthorJ.K.Rowling(youknowwhatshe’swritten)becameBritain’sfirstself-madefemalebillionaire.That’sfivehundredtimesthesizeofatypicalLottojackpot.Andtherearealotfewerthan14millionwritersinBritain.

ForgettheLottery.Writeabook.

TheCaseoftheFace-DownAcesFromtheMemoirsofDrWatsup

Mydetectivefriendsuddenlystoppedfiringhisrevolveratthechimneybreast,havinginscribeddottedversionsofthelettersVIGTOintheplaster.“Whatisit,Watsup?”saidheinanirritatedtone.

Isurfacedfrommyreverie.“I’msorry,Soames.WasIdisturbingyou?”“Icouldseeyouthinking,Watsup.Thewayyoupurseyourlips,andtugat

yourearwhenyoubelievenooneiswatching.Itismostdistracting.Onebulletwentawry,andnowtheClookslikeaG.”

“Iwasthinkingaboutthatnewstagemagicchap,”Isaid.“Um—”“TheGreatWhodunni.”“That’sthecove,yes.Cleverblighter.Wenttoseehisshowlastweek.Did

themostamazingcardtrick,beenwonderingaboutiteversince.Firsthetookapackofcardsanddealtthetopsixteenfacedowninfourrowsoffour.Thenheturnedfourofthecardsfaceup.Hecalledforavolunteerfromtheaudience,soofcourseIputmyhandup,butforsomereasonhechoseanattractiveyoungladyinstead.NameofHelena…well,anywayhetoldhertorepeatedly‘foldup’thesquareofcards,likefoldingasheetofstampsalongtheperforations,untilitendedupasasinglepileofsixteencards.”

“Shewasanaccomplice,”Soamesmuttered.“It’selementary.”“Idon’tthinkso,Soames.Thatwouldn’thavehelped.Theaudiencedecided

wherethefoldsoccurred.Forinstance,thefirstfoldcouldbealonganyofthethreehorizontallinesbetweenthecards,orthethreeverticallines—buttheaudiencecalledoutwhich.”

“Theaudiencewasanaccomplice,then.”Icouldtellhewasgettingintooneofhismoods.“Ichoseoneofthefolds

myself,Soames.”Thegreatmannoddeddistractedly.“Thenperhapsthetrickwasgenuine.In

whichcase—ah,yes,theConundrumoftheConcealedCupcakesspringstomind…Tell,me,Watsup:whenthecardshadbeenfoldedintoasinglepile,wasHelenainstructedtospreadthemoutonthetable?Withoutturninganyover?”

“Yes.”“Anddiditmiraculouslytranspirethateithertwelveofthecardswereface

downandfourfaceup,orfourwerefacedownandtwelvefaceup?”“Yes.Theformer.Andtheface-upcardswere—”“Thefouraces.Whatelse?Thewholethingisutterlytransparent.”“Butitcouldhavebeentheotherwayround,Soames,”Iprotested.“InwhichcaseHelenawouldhavebeentoldtotakethefourface-down

cards,andturnthemovertoreveal…”“Ah.Thefouraces.Isee.Butevenso,it’sanamazingpieceofmagic.Think

ofallthedifferentplacestheacescouldbe,andallofthewaystheaudiencecouldhavechosentofoldthecards!”

“Anamazingpieceofchicanery,Watsup.”Iallowedmyastonishmenttoshow.“Youmean—hefixedtheaudience’s

votes?Cleverpsychologicaltrickery?”“No,Watsup:hefixedthecards.GetmethatpackMrsSoapsudskeeps

underthehatstanddownstairsforherbridgeevening,andIwilldemonstrate.”Ihastenedtocarryouthisinstructions.

WhenIreappeared,puffingslightlyfrommyexertions,forIwasoutofcondition,Soamestookthecards.Hesortedoutthefouraces,andslidthembackintothepack,apparentlyatrandom.Afterdealingoutfourrowsoffourcardseach,heturnedfourofthecardsover,likethis:

Whodunni’sinitialarrangement

Thenheinstructedmetofoldthepackintoapile,followingthesameinstructionsthatWhodunnihadgiventoHelena.Thisdone,Ispreadthecardsout,andloandbehold,fourweretheoppositewayuptotheothertwelve.Andthosefourwere…theaces!

“Soames,”Icried,“thatistrulythemostamazingcardtrickIhaveeverseen!Idonowrealisethatyoumusthavechosenwheretoplacethefouraces,but

evenso,thenumberofdifferentwaysIcouldhavechosentofoldthecardsishuge!”

Soamesreloadedhisgun.“MydearWatsup,howmanytimeshaveItoldyounottoleaptounwarrantedconclusions.”

“Buttherereallyarethousandsofways,Soames!”Thedetectivegaveacurtnod.“Thatwasn’ttheconclusionIhadinmind,

Watsup.Doyoureallythinkthechoiceoffoldsmakestheslightestdifference?”Istruckmybrowwiththepalmofmyhand.“Youmean…itdoesn’t?”But

inreply,Soamesmerelyresumedhisattackonthechimneybreast.

HowdoesWhodunni’strickwork?Seepage265fortheanswer.

ConfusedParentsOneofthestrangestnamesforamathematicianisthatofHermannCa¨sarHannibalSchubert(1848-1911)whopioneeredenumerativegeometry,whichcountshowmanylinesorcurvesdefinedbyalgebraicequationssatisfyparticularconditions.Presumablyhisparentsexpectedgreatthingsoftheirson,butcouldn’tworkoutwhosesidetheywereon.

HermannCa¨sarHannibalSchubert

JigsawParadoxThetwotrianglesappeartohavethesamearea,namely13×5/2=32.5.Butonehasaholeinitsedge,sothediagramprovesthat31.5=32.5.What(ifanything)iswrong?

Seepage267fortheanswer.

Jigsawparadox

TheCatflapofFearFromtheMemoirsofDrWatsup

Hoovesskatedonthemuddyroad.Thecabscreechedroundacorner,narrowlymissingabarrowloadedwithpotatoes.Thecabbiewipedhisbrowwithadirtyrag.

“Cor,guv!ForamomentthereIthoughtwe’d’adourchips!”*“Driveon,man!There’saguineainitforyouifyouproceedwithallundue

haste!”Wearrivedatourdestination.Ileapedfromthecab,flingingcoinsatthe

driver,andrushedpastastartledMrsSoapsuds,upthestairs,andintoSoames’slodgings.Withoutknocking.

“Soames!It’sterrible!”Ipanted.“My—”“Yourcatshavebeenstolen.”“Catnipped,Soames!”“Surelyyoumean‘catnapped’?”“No,theywereluredawayusingabundleofcatniptiedtoapieceofstring.”“Howdoyouknowthat?”“Thecatnipperleftitbehind.”Soamesgavemeasharpstare.“Unusual.Notlikehim.Notlikehimatall.”“Him?”“Yes.He’sback.”Iwenttothewindow.“Soheis.Butthisishardlythetimeforroasted

chestnuts,Soames.”“Watsup,areyououtofyourmind?”“Theoldmanwhorunstheroastedchestnutstallacrossthestreet,”I

explained.“Hewasn’tthereyesterday,buttodayheis.Iassumethatistowhomyouarereferring.”

“Youassume,”Soamessaidscathingly.“Donotassume,Watsup.Examinetheevidenceanddeduce.”Irealisedthathewasnotmerelyspeakingingeneralities.Theremustbesomethingspecificthathewishedmetodeduce.

IpridemyselfinbeingunusuallysensitivetoSoames’smoods,andaftersomereflectionIrecalledthatafewdaysagoIhadcomeuponhimassemblingasmallarsenalofpistols,rifles,andhandgrenades.Nowitstruckmethatperhapsallwasnotwell.

Iputthishypothesistohim,andhenodded.“Itisasthoughaghostfromthepasthasrisenfromthegraveandissuckingthelifeoutoftheassembledmultitudesofhumanity,”hesaid.

“Isit?”Iasked.“Whatis,Soames?”“Afoulanddangerousfiend,theWellingtonofcrime.”“Doyounotmean‘Napoleon’?Itwouldseemmoreapt.TheDukewas

entirely—”“Hewearsrubberboots,”Soamesexplained.“Withanextremelycommon

treadpattern,todisguisehisfootprints.Hewearsgloves,toleavenofingerprints.Heisamasterofdisguise.Hecomesandgoeswithoutimpedimentthroughlockeddoors.Hehastheearofeverypolitician,theeyesofalltheirwives,andlongbeforethatfateddaywhenourpathsfirstcrossedhehadafingerineveryillicitpieinEngland.ButwithsuperhumaneffortItrackedhimdown,securedconvincingevidence,andbrokeuphisnetworkofcriminalgangs.Hefledthecountry,andIfoolishlythoughtthatwastheendofhim.Butnow,Ifindthathewasmerelylyinglow.Hehasreturned,andhehasresumedhisnefariousactivities.Andnowithasbecomepersonal.”

“Ofwhomdoyouspeak?”“Why,Mogiarty!ProfessorJimMogiarty,abrilliantbutflawed

mathematicianwhoturnedtotheDarkSide.Hebeganasamerecatburglar,beforeturninghisevilattentionstomoreprofitablecommodities.Notonlywillhestealanythingnotnaileddown:hewillalsostealnails,hammer,andfloorboards.Hehasdoggedmycareersince—”

“Soames:howcanacatburglardoganything?”“LikeIsaid,heisamasterofdisguise,Watsup.Dolisten.”“Andhowhashemanifestedhimself?”“Extortion,theft,murder,andkidnap.Andnow:catnip.Mogiartyis

revertingtohisoldways.”Hisexpressionwentgrimwithdetermination.“Neverfear,Watsup.Wewillrescueyourpets—”Iglaredathim—“yourfurryfelinecompanions.Youhavemyword.”

Ifinallythoughttoaskavitalquestion.“Soames?Howdidyouknowmycatsweremissing?”Silentlyheshowedmeatornenvelope.Insidewasascrapofpaperandasoggycatnipmouse.

“That’sDysplasia’smouse!”Istifledamanlysob.“What’sthepieceofpaperabout?”

Heshowedittome.Itread:

CSNSGISTCSTEEVTAOOHAGIAIEITNRETET

“Itisabitofajumble,Soames,butIseethewordsSTEEVandHAGIA.Er…doyouknowanyonecalledStepheninConstantinople?”

“No,Watsup!Itisacode.Ihavedecipheredit.”“How?”“Iobservedthatthereare33letters.Whatdoesthatsuggesttoyou,

Watsup?”“Er—therewasn’tmuchroomonthepaper.”“Watsup:33isequalto3×11,aproductoftwoprimes.Iinstantlythought

ofMogiarty’smathematicalpast.Anditoccurredtometorearrangethelettersina3×11rectangle.Likeso.”

C S N S G I S T C S TE E V T A O O H A G IA I E I T N R E T E T

Hebeamedwithpride;Icouldnotunderstandwhy.Itwasstillgibberish.“Readdownthecolumnsinorder,Watsup!”“CEASEINVESTIGATIONSORTHECATSGETIT.Ohdear!”Iwas

tremblingnowfromheadtotoe.“WhyisMogiartydoingsuchaterriblethingtoinnocentcreatures?”

“Heissendingusamessage.”“Thatmuchisclear.”“No,Imeantitmetaphorically.”“Ah.Hashedemandedaransom?”“No.Ibelieveittobeatest.Isuspectthatthiscrimeismerelyatrialrunfor

moreoutrageousones.Heistoyingwithuslikeacatwithamouse.”Istifledanothersob.“Whatcanwedo?”“Thegameisafoot,andwemustgetaheadofthegame,soasnottobetaken

aback.Alreadymytrustedinformantshavelocatedyourcatsinaperfectlyordinarylookinghouse—ironically,inBarking.Actuallyitisequippedwithconcealedmantraps,steeldoors,bulletproofwindows,andalarmsofseveralkinds.Thereisnopossibilityofusmountingaclandestinebreak-in.”

Iputmyservicerevolverbackinmypocket.“Apity.”“However,Mogiartyhasmadeamistake.Thereisaboarded-upcatflap.We

maybeabletorestoreitsfunctionandenticeyourcatsthroughit.”“Yes!”Icried.“Ihaveit!Wecantemptthemwiththeirfavouritetreats.

Aneurysmlikesartichokes,Borborygmusiscrazyaboutbananabread,Cirrhosiscanneverresistacreambun,andDysplasia’sdownfallisdumplings!”

“Dumplings,”saidhe.“Nevermind.Alittlebrainwork,somecrucialinformation,andyousee?Weprogress.Wecanemploytheseitemstoenticethecatstocomeoutthroughthecatflap.”

“Ihavesubstantialstocksofthenecessarycomestiblesathome,”Itoldhim.“Iwillgetthem.”

“Thatwillindeedbeuseful,Watsup,whenthetimecomes.Butthereisaproblem.Wemustpresentthedelicaciesinthecorrectorder,becausethecatsmustnotbepermittedtofight.”

“Ofcourse.Itcouldcauseinjury.”“No,becauseMogiartyhasfilledthebasementwithhighexplosive,and

riggedittodetonateiftheanimalsfight.”“What!Why?”“Becausehehasreasontobelievethatanyattempttorescuethecreatures

willprecipitateafelinealtercation.Heisusingtheanimalsthemselvesasawarningsystem.Typically,heignoresthevileconsequencesofhisbloodymachinations.AsIsaid,heissendingusamessage:thathewillstopatnothing.”

“Isee.”“Yousee,Watsup,butyoudonotobserve.Observationbeginswith

enquiries,whichprovideacontextfordeduction.Inowenquire.Inwhichcircumstancesdoyourcatsfight?Beprecise,forsuccessorfailuredependsonit.”

“Onlywhentheyareindoors,”Ireplied,afterduereflection.“Thenthehousemaygoupatanymoment!”“No:mycatsareentirelypeaceableprovidedcertaincombinationscanbe

prevented.”Iwrotedownalistofconditions:

IfCirrhosisandAneurysmarebothindoors,theyfightunlessDysplasiaispresent.IfDysplasiaandBorborygmusarebothindoors,theyfightunlessAneurysmispresent.IfAneurysmandDysplasiaarebothindoors,theyfightunlessBorborygmusorCirrhosis(orboth)arepresent.IfCirrhosisandDysplasiaarebothindoors,theyfightunlessBorborygmusorAneurysm(orboth)arepresent.IfAneurysmorBorborygmusareindoorsalone,theywon’tgooutatall.

HowcanSoamesandWatsupenticethecatsoutwithoutcausinganexplosion?Onlyonecatatatimecanusethecatflap.Ignoretrivialmoveswhereacatgoesoutandissentstraightbackin.Ifnecessary,acatcanbeshovedbackthrough

thecatflapaspartoftheprocess.

Seepage267fortheanswer.

Footnote

*Thisisnotananachronism:JosephMalinopenedthefirstfishandchipshopinLondonin1860.ThedelicacywasintroducedtoBritaininthesixteenthcenturybyJewishrefugeesofSpanishandPortugueseorigin,underthenamepescatofrito:deep-friedfish.Thechipswereaddedlater.

PancakeNumbersHereisagenuinemathematicalmystery—asimpleproblemwhoseansweriscurrentlyaselusiveasthecriminalmastermindMogiarty.

Youaregivenastackofcircularpancakes,allofdifferentsizes.Yourtaskistorearrangetheminorder,fromthelargestatthebottomtothesmallestatthetop.Theonlychangeyouareallowedtomakeistoinsertaspatulaunderneathsomepancakeinthestack,anduseittopickupthepancakesaboveitandfliptheentirepileover.Youcanrepeatthisoperationasmanytimesasyoulike,choosingwheretoplacethespatulaasyouwish.

Here’sanexamplewithfourpancakes.Ittakesthreeflipstogettheminorder.

Flippingastack

Herearesomequestionsforyou.1Caneverystackoffourpancakesbeputinorderusingatmostthreeflips?2Ifnot,whatisthesmallestnumberofflipsthatwillputanystackoffourpancakesinorder?3DefinethenthpancakenumberPntobethesmallestnumberofflipsthatwillputanystackofnpancakesinorder.ProvethatPnisalwaysfinite.Thatis,everystackcanbeputintotherightorderusingafinitenumberofflips.4FindPnforn=1,2,3,4,5.I’vestoppedatn=5becausetherearealready120differentstackstoconsider,andtobehonestthat’sanawfullotofwork.

Seepage269fortheanswers,andwhatelseisknown.

TheSoupPlateTrickContinuingthecookerytheme,thereisacurioustrickthatyoucanperformusingasoupplate,orsimilarobject.Startbybalancingtheplateonyourhand,inthemannerofawaiterservingdinner.Nowexplainthatyouwillaccomplishtheamazingfeatoftwistingyourarmthroughacompleteturnwhilekeepingtheplatehorizontalthroughout.

Todothis,firstrotateyourarminwards,tuckingtheplateunderyourarmpit.Continuetomovetheplateinacircle,butnowraiseyourarmaboveyourhead.Everythingreturnsnaturallytothestartingposition,andtheplatedoesn’tfalloffeventhoughyouarenotgrippingit.

Youcanfindvideosofthe(soup)platetrickontheInternet,forexampleat

http://www.youtube.com/watch?v=Rzt_byhgujg

whereitisreferredtoastheBalinesecuptrick,inreferencetoBalinesedanceusingacupfullofliquidinsteadofaplate.AsimilarPhilippinedanceusingwineglasses(twoperdancer,oneineachhand)canbeseenonYouTubeat

http://www.youtube.com/watch?v=mOO_IQznZCQ

Thismayseemafairlytrivialmanoeuvre,butithasdeepmathematicalconnections.Inparticular,ithelpsparticlephysicistsunderstandoneofthecuriousfeaturesofthequantumpropertyknownasspin.Quantumparticlesdon’treallyspin,likeaballbeingtwirledonajuggler’sfinger,butthereisanumber,calledspin,whichinacertainsensehasasimilareffect.Spinscanbepositiveornegative,analogoustoclockwiseandanticlockwise.Someparticleshavewholenumberspins:thesearecalledbosons(rememberthediscoveryoftheHiggsboson?).Others,morebizarrely,havehalf-integerspinslike or .Thesearecalledfermions.

Thehalvesarisebecauseofaverystrangephenomenon.Ifyoutakeaparticlewithspin1(oranyinteger)androtateitinspacethrough360°,itendsupinthesamestate.Butifyoutakeaparticlewithspin androtateitinspacethrough360°,itendsupwithspin– .Youhavetorotateitthrough720°,twofullturns,togetthespinbackwhereitbegan.

Themathematicalpointhereisthatthereisa‘transformationgroup’calledSU(2),whichdescribesspinandactsbytransformingquantumstates,andadifferentgroupSO(3)thatdescribesrotationsinspace.Thesearecloselyrelated,butnotidentical:everyrotationinSO(3)correspondstotwodistinct

transformationsinSU(2),oneofthembeingminustheother.Thisiscalledadoublecovering.It’sasifSU(2)wrapsroundSO(3),butitgoesroundtwice.Abitlikewrappingarubberbandtwiceroundabroomstick.

PhysicistsillustratethisideausingtheDiracstringtrick,namedforthegreatquantumphysicistPaulDirac.Theideatakesmanyforms;aparticularlysimpleoneusesaribbonwithoneendfixedandtheotherattachedtoarotor,whichfloatsinmid-air.Theribbonisshapedlikeaquestionmark.Afterarotationof360°theribbonhasnotreturnedtoitsoriginalposition,buttothatpositionrotatedthrough180°.Asecondfullrotationoftherotorto720°doesnottwisttheribbon,butputsitbackwhereitstarted.Thewaytheribbonmovesisessentiallythesameasthatofthearmholdingthesoupplate,exceptthattheplatemovesaroundabit.Anastronautfloatinginzerogravitycoulddothesamemovementswithafixedplate,whilekeepinghisbodypointingthesamedirectionatalltimes.

TheDiracstringtrickusingaribbon.Numbersshowtheangleindegreesthroughwhichtherotorhasturned.

Acomputer-generatedmovieAironDiracStringsbyGeorgeFrancis,LouKauffmanandDanielSandin(graphicsbyChrisHartmanandJohnHart)at

http://www.evl.uic.edu/hypercomplex/html/dirac.html

showstherelationshipbetweentheDiracstringtrickandthePhilippinewinedance.

Thesameideacanbeusedtoconnectelectricalcurrenttoarotatingdevicesuchasawheel.Atfirstsightthere’saproblem:thewheelhastohoverunsupportedinmid-air,toallowtheribbontodisentangle.However,in1975D.A.Adamsdesignedandpatentedadevicethatusesgearstoallowtheribbontorotatecompletelyroundthewheelonallsides.It’stoocomplicatedtoexplainhere,butseeC.L.Stong,Theamateurscientist,ScientificAmerican(December1975)120–125.

MathematicalHaikuThehaikuisashortJapaneseverseform,traditionallycomprisingthreeseparatephrases(lines)thatuseatotalof17syllables.TheactualJapaneseworddoesn’tcorrespondexactlytotheEnglishconceptofasyllable,butthatworkswellenoughforhaikuinEnglish.Thestricttraditionalpatternusesfivesyllablesinthefirstandthirdphrasesandseveninthemiddleone.Asanexample,here’sahaikubyMatsuoBashō,(1644–1694),inwhichboththeoriginal(omitted)andthetranslationhavethatformat:

Attheageoldpondafrogleapsintowateradeepresonance.

Inthesedecadentmoderntimes,the5-7-5patternisoftenrelaxed,withvariationssuchas6-5-6beingpermitted.Infact,thetotalof17syllablescanalsobechanged.Themostimportantfeatureisnotthepreciseform,buttheemotionalcontent,whichrequirespresentingtwodistinctbutlinkedimages.

Thesimpleformatofthehaikuhasadefinitemathematical‘feel’,andthereareinnumerablemathematicalhaiku.Forexample:

DanielMathews

RulerandcompassDegreeoffieldextensionMustbepoweroftwo.

JonathanAlperin

BeautifultheoremThebasiclemmaisfalseRejectthepaper.

JonathanRosenberg

Colloquiumtime.Lightsout,somebody’ssnoring.Mathissuchhardwork.

Accidentalhaikuoccurwhenwritersunintentionallyproduceasentencein

haikuformat.Forexample:

Andinthewestwardsky,Isawacurvedpalelinelikeavastnewmoon.

inTheTimeMachinebyH.G.Wells.OnethatAngelaBrettnoticed(amongmanyothers)inthePrincetonCompaniontoMathematicsis

Iseveryevennumbergreaterthanfourthesumoftwooddprimes?

TimPostonandIdedicatedour1977CatastropheTheoryandItsApplicationswithahaiku:

ToChristopherZeemanAtwhosefeetwesitOnwhoseshoulderswestand.

TheCaseoftheCrypticCartwheelFromtheMemoirsofDrWatsup

Soameswasrifflingthroughastackofnewspapers,seekingacrimethatwouldexercisehistalentssufficientlytobeworthinvestigating.Atthatmoment,Ihappenedtoglanceoutofthewindowandsawafamiliarfigurealightingfromahansomcab.“Why,Soames!”Icried.“Itis—”

“InspectorRoulade.Hewillbeheretorequestourassistance.”Therecameaknockonthedoor.IopenedittoseeMrsSoapsudsandthe

Inspector.“Soames!I’vecomeabout—”“TheDowninghamkidnappingcase.Yes,itdoeshavesomefeaturesof

interest.”HepassedRouladethenewspaper.“Asensationalistreport,MrSoames.Ill-informedspeculationsaboutthe

likelyfateoftheEarlofDowningham,andthesizeoftheransombeingdemanded.”

“Predictabilityofthepress,”saidSoames.“Yes.Althoughinthisinstanceitplaysintoourhandsbynotrevealing

certainkeyfactsthatmighthelpustoidentify—”“Thecriminal.Suchastheabsenceofanyransomdemand.”“HowonEarth—?”“Iftherehadbeenademand,itwouldbynowbepublicknowledge.Itisnot.

Evidentlythisisnoordinarykidnapping.WeshouldproceedwithallhastetoDowninghamHall.Which,ifmymemoryfailsmenot—anditneverdoes—isonUppinghamDown.”

“ThereisatrainfromKing’sCrosstoUppinghaminelevenminutes’time,”saidI,havinganticipatedhisdecisionandpulledacopyofBradshawfromthebookcase.

“Ifwepaythecabbieaguinea,wecanjustcatchit!”criedSoames.“Wecandiscussthecasewhilewetravel.”

UponourarrivalatDowninghamHall,theDukeofSouthmoreland—whoaccordingtothetime-honouredrulesofthenobilitywasthefatheroftheEarlofDowningham,whotookoneofhisfather’slessertitles—greetedusinperson,andquicklyconductedustothesceneoftheabduction,amuddypaddockoutsideabarn.

“Mysonvanishedsometimeduringthenight,”hestated,visiblyshaken.Soamesproducedhismagnifyingglassandcrawledaroundinthemudfor

severalminutes.Fromtimetotimehemutteredtohimself.Hetookoutatapemeasureandmadesomemeasurementsinacornerofthebarn.Thenherosetohisfeet.

“IhavealmostalloftheevidenceIneed,”hesaid.“WemustreturntoLondontofindthelastmissingpiece.”LeavingabaffledDukestandinginhisowndoorway,nexttoanequallybaffledInspector,wedidjustthat.

“But,Soames—”Ibeganwhenwehadboardedthetrain.“Didyounotnoticetheimpressionmadebythewheels?”hechallengedme.“Wheels?”“Thepolicehadtrampledallovertheevidence,asusual,butafewtraces

remained.EnoughformetodeterminethattheEarldepartedinafarmyardcart,oneofwhosewheelsfittedtightlyagainsttheendofthebarnwhereitmeetsahighwall.Atraceofmudonthewalltellsmethataspotontherimis8inchesfromthegroundand9inchesfromtheendofthebarn.Ifwecandeducethediameterofthewheel,thecasemaybeclosetoitssolution.”

Dataforwheel

“Maybe?”“Thatdependsontheanswer.Wemustalsobearinmindthatnocartwheel

hasadiameteroflessthan20inches.Letmesee…Ah,yes,itisasIsuspected.”UponourarrivalatKing’sCrossstationhecalledforaBakerStreetIrreducible—therewasalwaysoneofthelittlescampsnearby—anddispatchedhimtosendatelegramtoRoulade.

“Whatdoesitsay?”

“IttellshimwherethemissingEarlcanbefound.”“But—”“IknowofonlyonefarmintheneighbourhoodofDowninghamHallthat

hasacartwithwheelsoftheexactdiameterthatIhavecalculated,whichisdistinctivelylarge.IamconvincedthattheEarllefttheHallvoluntarilyundercoverofdarkness,usingahumblecarttoavoidattractingattention.Hewillbeattheplacewherethecartishabituallykept.”

NextmorningMrsSoapsudsbroughtatelegramfromtheInspector:EARLOFDSAFEANDWELLCONGRATULATIONSROULADE.

“SowheredidtheEarlgo?”Iaskedeagerly.“That,Watsup,isasecretwhosedisclosurewoulddestroythereputationsof

severalofthemostreveredfamiliesinEurope.ButIcantellyouthesizeofthewheel.”

Whatwasthewheel’sdiameter?Seepage271fortheanswer.

TwobyTwoTherearethousandsofNoah’sarkcartoons.Myfavouritehasabiologicaltheme.Thelastfewpairsofanimals—elephants,giraffes,monkeys—arebeingloadeduptherampintotheark.Noahisgrubbingaroundonthegroundonhandsandknees.Hiswifeisleaningoverthesideoftheark,yelling“Noah!Forgettheotheramoeba!”

There’samathematicalNoah’sarkjoke,wellwornbutperfectlycrafted.Asthefloodrecedes,Noahletsalltheanimalsloose,tellingthemtogoforth

andmultiply.Afterayearorso,hedecidestocheckuponthem.Therearebabyelephants,rabbits,goats,crocodiles,giraffes,hippos,andcassowarieseverywhere.Butthenhecomesacrossalonepairofsnakes,lookingdejected.

“What’stheproblem?”asksNoah.“Can’tmultiply,”saysoneofthesnakes.(BearinmindthatNoahisasortof

DrDolittlefigureandcantalktotheanimals.)Apassingchimpanzeeoverhearstheconversation.“Cutdownsometrees,

Noah.”Noahispuzzledbutdoesasthechimpsays.Afewmonths

laterhevisitsthesnakes,andnowtherearelittlesnakeseverywhereandeveryoneishappy.

“OK,howdidthathappen?”Noahasksthesnakes.“We’readders.Wecanonlymultiplyusinglogs.”

TheV-shapedGooseMysteryMigratingflocksofbirdsoftenflyinaV-shapedformation.V-shapedskeinsofgeeseareespeciallyfamiliar,andtheyoftencontaindozensorevenhundredsofbirds.Whatmakesthemadoptthisshape?

Researchershavelongsuggestedthatthisformationsavesenergybyavoidingbirdsgettingcaughtupintheturbulentwakeofthoseinfront,andrecentexperimentalandtheoreticalstudieshaveconfirmedthisgeneralviewpoint.Butthistheoryreliesonthebirdsbeingabletosensetheaircurrentsandadjusttheirflightaccordingly,anduntilrecentlyit’snotbeenclearthattheycandothis.

Analternativeexplanationisthattheflockhasaleader—theoneinfront—andeveryoneelsefollowstheleader.Perhapstheleaderisthebestnavigator,theonewhoknowswheretogo.Perhapsit’sjustwhicheverbirdfindsitselfatthefront.

BirdsinV-formation,flyingfromrighttoleft.Mostly.Where’sthebirdattoprightgoing?There’salwaysone…

Beforeproceedingtotheanswer,weneedtounderstandafewbasicfeaturesofbirdflight.Insteadyflight,abirdflapsitswingsinarepetitivecycle,down-beatfollowedbyup-beat.Itgainsliftfromthedown-beatasvorticesofairspinofffromtheedgesofthewings,anditusestheup-beattoreturnthewingtoitsoriginalposition,sothatthecyclecanrepeat.Thelengthofthecycleiscalledtheperiod.

Supposethattwobirdsareflyingusingcyclesofthesameperiod,whichisprettymuchwhathappensinamigratingflock.Althoughtheymoveinthesameway,theyneednotmakethesamemotionsatthesametime.Forexample,when

onebirdisproducingadown-beat,theothermaybeonanup-beat.Therelationbetweentheirtimingiscalledtherelativephase,anditisthefractionofacyclebetweenonebirdstartingitsdown-beatandtheotherbirdstartingitsowndown-beat.

ThankstosomeremarkabledetectiveworkbyStevenPortugalandhisteam,wenowknowthattheenergy-savingtheoryiscorrect,andthatthebirdscanindeedsensetheinvisibleaircurrentswellenoughtocarryitout.Thebigproblemforexperimentalstudiesisthatthebirdsyouaretryingtoobserverapidlydisappearfromview,alongwithanyattachedequipment.

Enterthebaldibis.OncethereweresomanybaldibisesthattheancientEgyptiansuseda

stylisedpictureastheirhieroglyphforakh,meaning‘toshine’.Todayonlyafewhundredsurvive,mainlyinMorocco.SoacaptivebreedingprogrammehasbeensetupatazooinVienna.Alotofeffortgoesintoteachingthebirdstofollowthecorrectmigrationroutes.Thisisdonebytrainingthemtofollowamicrolightaircraft,whichisflownalongpartsoftheroute—butalsoreturnstobase,alongwiththebirds.

Portugalrealisedthatitwouldbepossibletomakeextensivemeasurementsofthepositionsofthebirds,andhowtheymovetheirwings,fromtheaircraft.Insteadofthebirdsdisappearingoverthehorizon,theystayclosetotheequipment.Whathisteamfoundwasamazingandelegant.Eachbirdpositionsitselfbehindandslightlytothesideoftheoneinfront,anditadjuststherelativephaseofitswingflapssothatitridesontheupdraughtcreatedbythevortexspunoffbythebirdinfront.Thesecondbirdmustnotonlygetitswingtipintotherightplace,whichisrelativelytiny,butitmustalsoadjustthephaseofitsflaptoexploittheupdraughtefficiently.

Wingtipplacementandphaseadjustment.Greycurves:vorticesspunofffromwingtips.Arrowsshowrotationofvortex.

Atfirstsighttheseconsiderationswouldalsopermitazigzagformation,inwhicheachbirdfliestoonesideoftheoneinfront,butnotformingasingleVshape.(Ithasachoiceofflyingtotheleftortotheright.)However,thefirstbirdtobreaktheVshape(saybyflyingtotherightofthebirdaheadratherthanontheoutsideoftheVtoitsleft)wouldbedirectlybehindthebirdtwoplacesinfrontofit.Theairtherewouldbeturbulent,disturbedbythebirddirectlyinfront,soitwouldbemuchhardertoextractliftbyplacingthewingtipcorrectly.ThisproblemisavoidedbyflyingontheoutsideoftheVwheretheairisundisturbed.

Itwouldalsobepossibleforthebirdstoformasinglediagonalline,likeonearmoftheV.Butthiswouldleaveroomforbirdstojointheotherarm,closertotheleader.However,itiscommonforonearmoftheVtobelongerthantheother.

Whynotflyinamorecomplexzigzagformation,likethisorsomethingsimilarbutmorewiggly?

Inexperimentswithibises,ittookquiteawhileforjuvenilebirdstolearnhowtopositionthemselves.Inpracticeafewbirdsmaygetitwrong,andtheVshapeisseldomperfect.Nonetheless,thedetailedexperimentsshowconclusivelythatibisesarecapableofsensingtheairflowwellenoughtopositionthemselvesinorclosetothemostenergy-efficientlocationrelativetothebirdinfront.

Seepage272forfurtherinformation.

EelishMnemonicsThereareinnumerablemnemonicsforπ(seepages39–40).Mnemonicsforthatotherfamousmathematicalconstant,thebaseofnaturallogarithms

e=2.718281828459045235360287471352662497757…

arerarer.Amongthemaretwothatgivetenfigures:

Todisruptaplayroomiscommonlyapracticeofchildren.Itenablesanumskulltomemoriseaquantityofnumerals.

There’salsoa40-digitself-referentialmnemonicdevisedbyZeevBarel[Amnemonicfore,MathematicsMagazine68(1995)253],whichyoushouldcomparewiththedecimalexpansionabove.Itusesanexclamationmark‘!’inquotestorepresent0,anditgoeslikethis:

WepresentamnemonictomemoriseaconstantsoexcitingthatEulerexclaimed:‘!’whenfirstitwasfound,yes,loudly‘!’.Mystudentsperhapswillcomputee,usepowerorTaylorseries,aneasysummationformula,obvious,clear,elegant.

The‘easysummationformula’is

goingonforever.Now!denotesthefactorial

n!=n×(n–1)×…×3×2×1

SinceπmnemonicsarewritteninPilish[page39],areemnemonicswritteninEelish?

AmazingSquaresThereareinfinitelymanynaturalnumbersthatcanbeexpressedassumsofthreesquaresintwodifferentways:a2+b2+c2=d2+e2+f2.Butmoreisachievable.Anamazingexampleis

1237892+5619452+6428642=2428682+7619432+3237872

Therelationshipispreservedifwerepeatedlydeletetheleftmostdigit:

237892+619452+428642=428682+619432+23787237892+19452+28642=28682+19432+378727892+9452+8642=8682+9432+7872892+452+642=682+432+87292+52+42=82+32+72

Itisalsopreservedifwerepeatedlydeletetherightmostdigit:

123782+561942+642862=242862+761942+32378212372+56192+64282=24282+76192+323721232+5612+6422=2422+7612+3232122+562+642=242+762+32212+52+62=22+72+32

orifwedeletedigitsfrombothends:

23782+61942+42862=42862+61942+23782372+192+282=282+192+372

ThismathematicalmysterywassenttomebyMoloyDeandNirmalyaChattopadhyay,whoalsoexplainedthesimplebutcleverideainvolved.CanyouemulateHemlockSoamesandferretoutthesecret?

Seepage273fortheanswer.

TheThirty-SevenMysteryFromtheMemoirsofDrWatsup

“Howcurious!”Iremarked,thinkingaloud.“Manythingsarecurious,Watsup,”saidSoames,whomIhadthoughtto

havefallenasleepinhischair.“Whichcuriositydoyouhaveinmind?”“Itookthenumber123andrepeateditsixtimes,”Iexplained.“Getting123123123123123123,”Soamessaiddismissively.“Ah,yes,butIhaven’tfinished.”“Youmultipliedthatby37,nodoubt,”saidthegreatdetective,onceagain

confoundingmyexpectationthatIcouldtellhimanythingthathedidnotknowalready.

“Yes!Idid!AndwhatIgotwas—no,Soames,pleasedonotinterrupt—theanswer

4555555555555555551

withagreatmanyrepetitionsofthedigit5.”“Andthatiscurious?”“Undoubtedly.Whileonesuchcalculationmightbemerecoincidence,

somethingsimilarhappensifIuse234or345or456insteadof123.Look!”AndIshowedhimmyarithmetic:

234234234234234234×37=8666666666666666658345345345345345345×37=12777777777777777765456456456456456456×37=16888888888888888872

“Notonlythat:ifIrepeat123or234or345or456adifferentnumberoftimes,andmultiplyby37,Iagainobtainagreatmanyrepetitionsofthesamedigit,exceptneartheends.”

“Iaminclinedtothink,”murmuredSoames,“thatthepatterns123,234,345,andsoon,areirrelevant.Haveyoutriedothernumbers?”

“Itried124,andthatdidn’twork.Look!”

124124124124124124×37=4592592592592592588

“Digitsrepeatinblocksofthree,butIdon’tfindthatsurprising,sincetheydothesameinthefirstnumber.”

“Didyoutry486?”“No—well,since124fails,Ireallydon’tthink…Oh,verywell.”Ireturned

tomynotebookandwrotedownthecalculation.“Howcurious!”Isaidagain,havingdiscoveredtheanswer:

486486486486486486×37=17999999999999999982

Inspired,Itriedvariousthree-digitnumbersatrandom,writingthemdownsixtimesinarowandmultiplyingby37.Sometimestheresultincludedmanyrepetitionsofthesamedigit,butmoreoften,itdidnot.IshowedSoamesmyworking,andconfessed:“Iamflummoxed.”

“Themysterywillnodoubtresolveitself,”Soamesreplied,“ifyouconsiderthenumber111.”

Iwrotedown

111111111111111111×37=4111111111111111107

andstaredatit.Aftertwentyminuteshadpassed,Soamesgotupandglancedovermyshoulder.Heshookhisheadinamusement.“No,no,Watsup!Iwasnotsuggestingyoutryyourmethodonthenumber111!”

“Oh.Iassumed—”“HowmanytimeshaveItoldyou,Watsup:donotassumeanything!

Althoughthemysteryseemstoinvolvethenumber37,thatissomethingofasideissue.Iwassuggestingthatyoushouldcontemplatehowthenumber111relatesto37.”

Seepage273fortheanswer.

AverageSpeedBecauseofheavytraffic,abusdrivesfromEdinburghtoLondon,adistanceof400miles,in10hours,aspeedof40mph.Itmakesthereturnjourneyin8hours,aspeedof50mph.Whatisitsaveragespeedforthewholejourney?

Theobviousansweris45mph,thearithmeticmeanof40and50,obtainedbyaddingthemanddividingby2.However,thebusperformsthetotalroundtripof800milesin18hours,anaveragespeedof800/18= mph.

Howcome?

Seepage275fortheanswer.

FourCluelessPseudokuThecluelesspuzzleonpage41wascreatedbyGerardButters,FrederickHenle,JamesHenle,andColleenMcGaughey.ItisavariantofsudokuwhichIliketocallcluelesspseudoku.Herearefourmorecluelesspseudokumysteriestosolve.Therulesare:

Eachrowandeachcolumnmustcontaineachnumber1,2,3,…,nonceonly,wherenisthesizeofthesquare.Thenumbersineachoftheregionsoutlinedinthickblacklinesmusthavethesametotal.I’vewrittenitabovethepuzzletosaveyouthebotherofworkingitout.Eachsolutionisuniqueexceptforthefourthpuzzle,wheretherearetwosolutions,symmetricallyrelated.

Fourcluelesspseudokumysteries

Seepage276fortheanswersandfurtherreading.

SumsofCubesTriangularnumbers1,3,6,10,15,andsoon,aredefinedbyaddingconsecutivenumberstogether,startingfrom1:

1=11+2=31+2+3=61+2+3+4=101+2+3+4+5=15

andsoon.Thereisaformula

1+2+3+…+n=n(n+1)/2

andonewaytoproveitistowritethesumtwice,likethis:

1+2+3+4+55+4+3+2+1

andobservethatthenumbersinverticalcolumnsalladdtothesamething,here6.Sotwicethesumis6×5=30,andthesumis15.Ifyoudidthiswiththenumbersfrom1to100,itwouldworkinmuchthesameway:therewouldbe100columnseachaddingto101,sothesumofthefirst100numbersmustbehalfof100×101,whichis5,050.Moregenerally,ifweaddthefirstnnumberswegethalfofn(n+1),andthat’stheformula.

There’saformulaforsumsofsquares,butit’sabitmorecomplicated:

1+4+9+…+n2=n(n+1)(2n+1)/6

Butwhathappensforcubesisverystriking:

13=113+23=913+23+33=3613+23+33+43=10013+23+33+43+53=225

Theresultsarethesquaresofthecorrespondingtriangularnumbers.Whyshouldsumsofcubesgivesquares?Wecanfindtheformulaandprove

itthatway,butthere’saveryneatpictorialproofthat

13+23+33+…+n3=(1+2+3+…+n)2

withoutusinganyformulas.

Visualisingsumsofcubes

Thepictureshows1squareofside1,2ofside2(makinga2×2×2cube),3ofside3(a3×3×3cube),andsoon.Sothetotalareaisthesumofconsecutivecubes.Readingalongthetopedgewefind1+2+3+4+5,thesumofconsecutivenumbers.Buttheareaofasquareisthesquareofthelengthofitsside.Done!

Ifyouwantaformula,weknowthat(1+2+3+…+n)=(n(n+1)/2,sosquaringgives13+23+33+…+n3=n2(n+1)2/4.

ThePuzzleofthePurloinedPapersFromtheMemoirsofDrWatsup

Soamespassedmeanenvelope,andhelduptheletterthatithadenclosed.“Atestofyourpowersofobservation,Watsup.Whodoyouthinksentme

this?”Iheldittothelight,lookedatthepostmarkandstamp,sniffed,examinedthe

gluewhereithadbeensealed.“Thesenderisawoman,”Isaid.“Unmarriedbutnotyetcondemnedtospinsterhood,andactivelyseekingahusband.Sheisfrightened,butcourageous.”Ipaused,andfurtherinspirationstruck.“Herfinancesarestrained,butnotyetdisastrous.”

“Verygood,”saidhe.“Iseeyouhaveabsorbedsomeofmymethods.”“Idomybest,”saidI.“Explainwhatledyoutothesedeductions.”Iorderedmythoughts.“Theenvelopeispink,anditbearsdistincttracesof

perfume.NuitsdePlaisir,ifIamnotmistaken,formyfriendBeatrixoftenwearsthesame.Itistoodaringforamarriedwoman,notdaringenoughforayoungone.Thatshewearsperfumeatallimpliessheisactivelyseekingmaleattention.Tracesofcosmeticsontheflapconfirmthis.Butthegluehadbeenlickedonlypartially,suggestingthathermouthwasdrywhenshesealedtheenvelope,andadrymouthisasignoffear.Sincesheneverthelesscompletedthetaskandpostedthelettertoyou,sheisstillabletofunctionrationally,underseverestress,asignofcourage.

“Finally,thestampshowssignsofhavingbeensteamedoffanotherenvelopeandbeenreused—abentcorner,tracesofapreviouspostmark.Thisindicatesafrugalattitude.However,shecanaffordperfume,sosheisnotyetonpoverty’sdoorstep.”

Henoddedthoughtfully,andImentallypreenedmyself.“Thereareafewsignsyoumissed,”hesaidquietly,“whichputthematterin

afreshlight.Theshapeandsizeoftheenveloperevealitasgovernmentissue,nottobefoundinanyhigh-streetstationer’s.Ireferyoutomymonographonstationeryanditscharacteristicdimensions.Theinkusedtowritetheaddressisastrangeshadeofdarkbrown,unavailablecommerciallybutprovidedinbulktocertaindepartmentsinWhitehall.”

“Ah!Thenhercurrentbeauisacivilservantandsheborrowedboth

envelopeandinkfromhim.”“Asensibletheory,”saidhe.“Totallyincorrect,ofcourse,buteminently

sensibleandconsistentwithmuchoftheevidence.However,infactthisletterisfrommybrotherSpycraft.”

Iwasshockedrigid.“Youhaveabrother?”Soameshadnevertalkedofhisfamily.

“Oh,haveInotmentionedhim?Howveryremissofme.”“Howdoyouknowhewrotetheletter?”“Hesignedhisnametoit.”“Oh.Butwhatoftheotherclues?”“Spycraft’slittlejoke.Butwemustmakehaste,forwearetomeethim

forthwithattheDiophantusClub.Giveanurchinsixpencetocallahansom,andIwilltellyoumoreaswemakeourwaythere.”

AsweclatteredalongPortlandPlace,heexplainedthathisbrotherwasaretiredexpertonprimenumbers,whodidoccasionalfreelanceworkforHerMajesty’sGovernment.Herefusedtobedrawnonthenatureofthework,sayingonlythatitwashighlyconfidentialandpoliticallysensitive.

UponarrivingattheDiophantusClub,wewereusheredintotheVisitor’sLounge,whereagentlemanwaswaitinginacomfortablearmchair.Myimmediateimpressionwasoneoflanguidcorpulence,butitconcealedasharpnessofmindandalertnessofbodythatbeliedthisinitialassessment.

Soamesintroducedus.“Youoftenfindmyowndeductiveabilitiesastonishing,Watsup,”hesaid,

“butSpycraftputsmetoshame.”“Thereisoneareawhereyourabilitiesexceedmyown,”hisbrother

contradicted.“Namely,logicalconundrumsinwhichthepreciseconditionsarefluid.IfindthatIhavenobasisfromwhichtoattackthequestion.Whencemynote.”

“ItakeitthatyouhavenoobjectiontoDrWatsupbeingtoldall?”“HisservicerecordinAl-Jebraistanisimpeccable.Hemustbeswornto

secrecy,buthiswordwillbesufficient.”Soamesgavehisbrotherasharplook.“It’snotlikeyoutoacceptsomeone’s

word.”“ItwillbesufficientwhenIinformhimoftheconsequencesofbreakingit.”Idulyswore,andwegotdowntobusiness.“Animportantdocumentwasaccidentallymislaid,andthenstolen,”Spycraft

said.“ItisessentialtothesecurityoftheBritishEmpirethatitberecoveredwithoutdelay.Ifitgetsintothehandsofourenemies,careerswillberuinedandpartsoftheEmpiremayfall.Fortunately,alocalconstablecaughtaglimpseof

thethief,enoughtonarrowitdowntopreciselyoneoffourmen.”“Pettythieves?”“No,allfouraregentlemenofhighrepute.AdmiralArbuthnot,Bishop

Burlington,CaptainCharlesworth,andDoctorDashingham.”Soamessatboltupright.“Mogiartyhasahandinthis,then.”Notfollowinghisreasoning,Iaskedhimtoexplain.“Allfourarespies,Watsup.WorkingforMogiarty.”“Then…Spycraftmustbeengagedincounter-espionage!”Icried.“Yes.”Heglancedathisbrother.“Butyoudidnothearthatfromme.”“Havethesetraitorsbeenquestioned?”Iasked.Spycrafthandedmeadossier,andIreaditaloudforSoames’sbenefit.

“UnderinterrogationArbuthnotsaid‘Burlingtondidit.’Burlingtonsaid‘Arbuthnotislying.’Charlesworthsaid‘ItwasnotI.’Dashinghamsaid‘Arbuthnotdidit.’Thatisall.”

“Notquiteall.Weknowfromanothersourcethatexactlyoneofthemwastellingthetruth.”

“YouhaveaninformerinMogiarty’sinnercircle,Spycraft?”“Wehadaninformer,Hemlock.Hewasgarrottedwithhisownnecktie

beforehecouldtellustheactualname.Verysad—itwasanOldEtoniantie,totallyruined.However,allisnotlost.Ifwecandeducewhowasthethief,wecanobtainasearchwarrantandrecoverthedocument.Allfourmenarebeingwatched;theywillhavenoopportunitytopassthedocumenttoMogiarty.Butourhandsaretied;wemuststicktotheletterofthelaw.Moreover,ifweraidthewrongpremises,Mogiarty’slawyerswillpublicisethemistakeandcauseirreparabledamage.”

Whichmanwasthethief?Seepage276fortheanswer.

MasterofAllHeSurveysAfarmerwantedtoencloseaslargeanareaoffieldaspossible,usingtheshortestpossiblefence.Perhapsunwisely,hecalledthelocaluniversity,whosentanengineer,aphysicist,andamathematiciantoadvisehim.

Theengineerbuiltacircularfence,sayingitwasthemostefficientshape.Thephysicistbuiltastraightlinesolongthatyoucouldn’t

seetheends,andtoldthefarmerthattoallintentsandpurposesitwentrightroundtheEarth,sohehadfencedinhalftheplanet.

Themathematicianbuiltatinycircularfencearoundhimselfandsaid“Ideclaremyselftobeontheoutside.”

AnotherNumberCuriosity1×8+1=912×8+2=98123×8+3=9871234×8+4=987612345×8+5=98765

So,allyoubuddingHemlockSoameses:whatcomesnextandwhendoesthepatternstop?

Seepage278fortheanswers.

TheOpaqueSquareProblemSpeakingoffences:whatistheshortestfencethatwillblockeverylineofsightacrossasquarefield?Thatis,afencethatmeetseverystraightlinethatintersectsthefield.ThisistheOpaqueSquareProblem:thenameindicatesthatyoucan’tseethroughit.ThequestiongoesbacktoStefanMazurkiewiczin1916,whoaskeditforanyshape,notjustasquare.Itremainsbaffling,butsomeprogresshasbeenmade.

Supposethesideofthefieldisoneunit.Thenafenceroundallfoursideswouldcertainlywork,andthelengthwouldbe4.However,wecouldcutoutoneofthesidesandstillhaveanopaquefence,reducingtheanswerto3.Thisistheshortestfenceformedbyasinglepolygonalline.Butifweallowfencesusingseverallines,ashorterpossibilityquicklyspringstomind:thetwodiagonalsofthefield,totallength ,approximately.

Canwedobetterstill?Onegeneralfactisclear:anopaquefencecontainedentirelywithinthefieldmustincludeallfourcornersofthesquare.Ifsomecornerwerenotincluded,therewouldbealineintersectingthesquareonlyatthatcorner(cuttingdiagonallythroughitfromoutside)andthatwouldmissthefence.Evenonesuchlinebreakstheconditionsoftheproblem.

Anyfencethatincludesallfourcornersandjoinsthemtogethermustbeopaque,becauseanylinethatintersectsthesquaremusteitherpassthroughacornerorseparatetwoofthem.Thenanyconnectingfencemustcrossthatline.Isthepairofdiagonalstheshortestsuchfence?No.Theshortestfencethatconnectsallfourcorners,calledaSteinertree,haslength =2.732,approximately.Thelinesmeetatanglesof120°.

Itturnsoutthateventhisfenceisnottheshortestopaqueone.Thereisadisconnectedfence,inwhichonepartblockslinesofsightthroughthegap,oflength .Itiswidelybelieved,butnotyetproved,thatthisistheshortestopaquefence.BerndKawohlhasprovedthatthisistheshortestfencethathasexactlytwoconnectedpieces.OneistheSteinertreelinkingthreecorners,threelinesthatmeetat120°theotheristheshorteststraightlinebetweenthecentreandthefourthcorner.

Opaquefencesforasquare.Fromlefttoright:lengthsare4,3,2.828,2.732,and2.639.

Wedon’tevenknowforsurethatthereisashortestopaquefence.Orthatifthereisone,itmustbeentirelywithinthesquare.VanceFaberandJanMycielskihaveprovedthatforanygivenfinitenumberofpieces,atleastoneshortestopaquefenceexists.(Forallweknow,theremightbeseveral.)Thetechnicalproblemhere,whichiscurrentlyunsolved,isthepossibilitythatthemorepiecesyoupermit,theshorterthefencecanbe.Itwouldthenbepossibletofindaseriesoffenceswithever-shorterlengths,butnofencethatisshorterthanallofthese.Alternatively,afencecomposedofinfinitelymanydisconnectedpiecesmightbetheshortest.

OpaquePolygonsandCirclesAstandardmathematician’strick,whenyoucan’tsolveaproblem,istogeneraliseit:considerarangeofsimilarbutmorecomplicatedproblems.Thismightseemastupididea:howcanmakingthequestionharderhelpyoutosolveit?Butthemoreexamplesyouhavetothinkabout,thebetteryourchancesofspottingsomeinterestingcommonfeaturethatcrackstheproblem.Itdoesn’talwayswork,andsofarithasn’tdonehere,butoccasionallyitdoes.

OnewaytogeneralisetheOpaqueSquareProblemistochangetheshape.Replacethesquarebyarectangle,orapolygonwithmoresides,acircle,anellipse—thepossibilitiesareendless.

Mathematicianshavemainlyconcentratedontwogeneralisations:regularpolygonsandcircles.TheshortestknownopaquefencefortheequilateraltriangleisaSteinertreejoiningeachcornertothecentrebyastraightline.Thereisageneralconstructionthatgivestheshortestknownopaquefencesforregularpolygonswithanoddnumberofsides,andasimilarbutdifferentoneforanevennumberofsides.

Shortestknownopaquefencesforregularpolygons.Fromlefttoright:equilateraltriangle;odd-sidedregularpolygon;even-sidedregularpolygon.

Whataboutanopaquecircle?Ifthefencehastostaywithintheshape,theobviousansweristheperimeterofthecircle.Ifit’saunitcircle,thishaslength2π=6.282.Ifpartoftheperimeterismissing,youneedextrabitsoffenceinsidethecircletoblockpathsthatcrossthismissingsegment,anditgetscomplicated.Intuitively,acirclecanbethoughtofasaregularpolygonwithinfinitelymanyinfinitelyshortsides.Basedonthisidea,Kawohlhasprovedthataconstructionlikethatforregularpolygons,butusinganinfinitenumberofpieces,givesanopaquefenceoftotallengthπ+2=5.141,whichissmallerthan2π.ButthereisashorterU-shapedopaquefenceifsomeofitcanlieoutsidethecircle.Thisalsohaslengthπ+2,andisconjecturedtobetheshortestpossible,andithasbeenprovedtobesoforfencesformedbyasinglecurvewithnobranchpoints.

Opaquefencesforacircle.Left:Obviousbutnotthebest.Right:Ashorteropaquefencethatgoesoutsidethecircle.

Theproblemhasalsobeenextendedtothreedimensions:nowthefencehastobeasurface,orsomethingmorecomplicated.Thebestknownopaquefenceforacubeisformedfromseveralcurvedpieces.

Thebestknownopaquefenceforacube.

πr2?No,pieareround.Chocolatearesquared.

TheSignofOneFromtheMemoirsofDrWatsup

“Soames!Here’saprettypuzzle.Itmightinterestyou.”HemlockSoamesputdownhisclarinet,uponwhichhehadbeenplayinga

Bolivianfuneraldirge.“Idoubtit,Watsup.”Hehadbeeninthismelancholymoodforsomeweeks,andIwasdeterminedtokickhimoutofit.

“Theproblemistoexpresstheintegers1,2,3,andsoon,usingatmost—”“Four4’s,”saidSoames.“Iknowitwell,Watsup.*”Idecidednottolethislackofenthusiasmdauntme.“Withbasicarithmetical

symbolsitispossibletoreach22.Squarerootsincreasethislimitto30.Factorialsraiseitto112;powersto156—”

“Andsubfactorialsto877,”Soamesfinished.“Itisanoldpuzzle,andonethathasbeenwrungdry.”

“What’sasubfactorial,Soames?”Iasked,buthehadalreadyburiedhisnoseinyesterday’sDailyWail.

Afteramoment,hereappeared.“Mindyou,therearemanypossiblevariations.Theuseof4allowsconsiderablefreedom,andseveralusefulnumberscanbecreatedusingjustone4.Suchas =2and4!=24.”

“Whatdoestheexclamationmarkmean,”Iasked.“Factorial.Forinstance,4!=4×3×2×1,andsoon.Which,asIsaid,is

24.”“Oh.”“Theseextranumberscomefreeofcharge,andsomakethepuzzleeasier.

ButIwonder…”Hisvoicetrailedoff.

“Wonderwhat,Soames?”“Iwonderhowfaronecangetusingfourones.”SilentlyIrejoiced,forhisinterestwasclearlypiqued.“Yes,Isee,”Isaid.

“Now =1and1!=1,sononewnumbersarise‘forfree’.Whichmakestheproblemharder,andperhapsmoreworthyofourattention.”

Hegrunted,andIhastenedtopresshomemyslightadvantage.ThebestwaytointerestSoamesinaproblemwastoattemptit,andfail.

“Iseethat

1=1×1×1×1

and

2=(1+1)×1×13=(1+1+1)×14=1+1+1+1

Butanexpressionfor5escapesme.”Soamesraisedoneeyebrow.“Youmightconsider

5=(1/.1)/(1+1)

wherethedotisthedecimalpoint.”“Oh,that’sratherclever!”Icried,butSoamesmerelysnorted.“Sowhat

about6?”Icontinued.“Icanseehowtodoitusingfactorials:

6=(1+1+1)!×1

Ireallyonlyneedthree1’s,butanysparescanalwaysbegotridofbymultiplyingby1.”

“Elementary,”hemuttered.“Haveyouconsidered

Watsup?Or,forthatmatter,

ifyouinsistonemployingfactorials.Youmaymultiplyby1×1or1/1,oradd1–1,touseallfour1’s,ofcourse.”

Istaredattheformula.“Irecognisethedecimalpoint,Soames,butwhatistheextra ?”

“Recurring,”Soamesrepliedwearily.“Noughtpointonerecurringisequalto0.111111…goingonforever.Ofcoursetheinitialzeromaybeomitted.Theinfiniterecurringdecimalequals1/9exactly.Dividingthatinto1gives9,whosesquarerootis3—”

“Andthen3+3=6,”Iyelledexcitedly.“And,ofcourse,

7=(1+1+1)!+1

iseasywithoutsquareroots.But8isakettleofwormsofadifferentcolour—”“Dopayattention,”saidSoames.

8=1/. –1×19=1/. +1–1

“Ah!Yes!Andthen

10=1/. +1×111=1/. +1+1

and…”“Youareusingup1’sprodigiously,”saidSoames.“Itisbesttosavethem

forlater.”Andhewrotedown

10=1/.111=11

adding,“Notetheabsenceofthe‘recurring’symbol,Watsup.Thistimeitisjusttheordinarydecimal.1.Oh,andyoumustmultiplybothby1×1touseupthespare1’s,ordosoinoneoftheotherwaysIremarkedupon.Butlater,youcanomitthosetwo1’sandputthemtogoodpurpose.”

“Yes!Youmean,like

andsoon?”AflickerofasmilecrossedSoames’scountenance.“Youhaveit,Watsup!”“Butwhatabout15?”Iasked.“Trivial,”hesighed,andwrotedown

TowhichItriumphantlyappended

andSoamesnoddedapprovingly.“Nowitstartstobecomeinteresting,”heremarked.“Whatof23,Iwonder?”

“I’vegotit,Soames!”Icried:

Bearinginmind,”Iclarified,“that4!=24,asyousosagelyremarked.Thisisfun,Soames!ThoughforthelifeofmeIcan’tmanagetoexpress26.”

“Well…”hebegan,andpaused.“Stumped,you,hasit?”“Notintheleast.Iwasmerelywonderingwhetheritisnecessaryto

introduceanewsymbol.Itwillcertainlymakelifeeasier.Watsup,areyouawareofthefloorandceilingfunctions?”

Mygazeinadvertentlywenttomyfeet,andthenupabovemyhead,butnoinspirationstruck.

“Iseeyoudonot,”saidSoames.HowdoesheknowwhatI’mthinking?Ithought.It’s—

“Uncanny…yes,isitnot?Ireadyoulikeabook,Watsup.PossiblyMotherGoose.Now,thesefunctionsare

[x]=thegreatestintegerlessthanorequaltox(floor)[x]=thesmallestintegergreaterthanorequaltox(ceiling)

andyouwillfindthemindispensableinallpuzzlesofthissort.”“Excellent,Soames.ThoughIadmit,Ifailtosee…““Theidea,Watsup,isthatbytheirmeanswecanexpressusefulsmall

numbersusingonlytwo1’s.Forexample,

isanotherwaytorepresent3withtwo1’s,and

isnew.”Seeingmybewilderment,headded:“Youappreciatethat,whoseflooris3andceilingis4.”

“Yes…”Isaiddoubtfully.“Thenweadvance,because

Nottomentionvariousalternatives.”Thousandsofincoherentthoughtssurgedthroughmybrain.Onestoodout.

“Why,Soames,I’vejustrealisedthat

because ,whoseceilingis5.SoIcannowmake29and30!”BywhichImeant30,notfactorial30,youunderstand.Punctuationissuchapain.

WatsupandSoamesinvestigatedthepuzzlemuchfurther,andwewillseewhattheydiscoveredlater.Butbeforecontinuingthestory,youmightliketoseehowfaryoucangetonyourown.Startingwith31.

TheSignofOnecontinuesonpage105.

Footnote

*W.W.RouseBall,MathematicalRecreationsandEssays(11thedition),Macmillan,London1939.

ProgressonPrimeGapsRecallthatawholenumberiscompositeifitcanbeobtainedbymultiplyingtwosmallerwholenumberstogether,andprimeifitcan’tbeobtainedbymultiplyingtwosmallerwholenumberstogetheranditisgreaterthan1.Thenumber1isexceptional:afewcenturiesagoitwasconsideredtobeprime,butthisconventionstopsprimefactorsbeingunique.Forexample,6=2×3=1×2×3=1×1×2×3,andsoon.Nowadays,forthisreasonandothers,1isconsideredtobespecial.Itisneitherprimenorcomposite,butaunit:awholenumberxsuchthat1/xisalsoawholenumber.Indeed,itistheonlypositiveunit.

Thefirstfewprimesare

23571113171923293137

Thereareinfinitelymanyofthem,scatteredinanirregularmannerthroughoutthewholenumbers.Theprimenumbershavelongbeenahugesourceofinspiration,andmanyoftheirmysterieshavebeensolvedovertheyears.Butmanyothersremainasopaqueastheyhaveeverbeen.

In2013numbertheoristsmadesuddenandunexpectedprogressontwoofthegreatmysteriesaboutprimenumbers.Thefirstconcernedthegapsbetweensuccessiveprimes,andI’lldescribethatnow.Thesecondfollowsimmediatelyafter.

Allprimesotherthan2areodd(sinceallevennumbersaremultiplesof2),soitisnotpossiblefortwoconsecutivenumbersexcept(2,3)bothtobeprime.However,itispossiblefortwonumbersthatdifferby2tobeprime:examplesare(3,5),(5,7),(11,13),(17,19),anditiseasytofindmore.Suchpairsarecalledtwinprimes.

Ithaslongbeenconjecturedthatthereareinfinitelymanytwinprimepairs,butthishasnotyetbeenproved.Untilrecentlyprogressonthisquestionwasminimal,butin2013YitangZhangstunnedthemathematicalworldbyannouncingthathecouldprovethatinfinitelymanypairsofprimesdifferbyatmost70million.HispaperhassincebeenacceptedforpublicationbytheleadingpuremathematicsjournalAnnalsofMathematics.Thismaysoundfeeblecomparedtothetwinprimeconjecture,butitwasthefirsttimeanyonehadshownthatinfinitelymanyprimesdifferbysomefixedamount.If70millioncouldsomehowbereducedto2,thiswouldresolvethetwinprimesconjecture.

Today’smathematiciansincreasinglyusetheInternettojoinforcesonproblems,andTerenceTaoorchestratedacollaborativeefforttoreducethefigureof70milliontosomethingmuchsmaller.HedidthiswithintheframeworkofthePolymathproject,asystemsetuptofacilitatethiskindofwork.AsmathematiciansgainedabetterunderstandingofZhang’smethods,thenumbertumbled.JamesMaynardreducedthefigureof70millionto600(indeedto12ifanotherconjecturecalledtheElliott–HalberstamConjectureisassumed,Seepage278).Bytheendof2013newideasbyMaynardhadreduceditto270.

Notyet2,butalotcloserthan70million.

TheOddGoldbachConjectureThesecondprimemysterytoberesolved(probably!)goesbackto1742,whentheGermanamateurmathematicianChristianGoldbachwrotealettertoLeonhardEulercontainingseveralobservationsaboutprimes.Onewas:“Everyintegergreaterthan2canbewrittenasthesumofthreeprimes.”EulerrecalledapreviousconversationinwhichGoldbachhadmadearelatedconjecture:“Everyevenintegeristhesumoftwoprimes.”

Withtheconventionthenprevailing,that1isprime,thisstatementimpliesthefirstone,becauseanynumbercanbewrittenaseithern+1orn+2whereniseven.Ifnisthesumoftwoprimes,theoriginalnumberisthesumofthreeprimes.Eulersaid“Iregard[thesecondstatement]asacompletelycertaintheorem,althoughIcannotproveit.”Thatprettymuchsumsupitsstatustoday.

Goldbach’slettertoEuler,statingthatifanumberisasumoftwoprimesthenitisasumofanynumberofprimes(uptothesizeofthenumberconcerned).Inthemarginistheconjecturethateverynumbergreaterthan2isasumofthreeprimes.Goldbachdefined1tobeprime,whichisnotthemodernconvention.

However,wenolongerdeem1tobeprime,asdiscussedonpage90.So

nowadayswesplitGoldbach’sconjecturesintotwodifferentones:

TheevenGoldbachconjecturestates:

Everyevenintegergreaterthan2isthesumoftwoprimes.

TheoddGoldbachconjectureis:

Everyoddintegergreaterthan5isthesumofthreeprimes.

Theevenconjectureimpliestheoddone,butnotconversely.Overtheyears,variousmathematiciansmadeprogressonthesequestions.

PerhapsthestrongestresultontheevenGoldbachconjectureisthatofChenJing-Run,whoprovedin1973thateverysufficientlylargeevenintegeristhesumofaprimeandasemiprime(eitheraprimeoraproductoftwoprimes).

In1995theFrenchmathematicianOlivierRamaréprovedthateveryevennumberisasumofatmostsixprimes,andeveryoddnumberisasumofatmostsevenprimes.TherewasagrowingbeliefamongtheexpertsthattheoddGoldbachconjecturewaswithinreach,andtheywereright:in2013HaraldHelfgottclaimedaproofusingrelatedmethods.Hisresultisstillbeingcheckedbyexperts,butseemstobeholdingupwellunderscrutiny.Itimpliesthateveryevennumberisthesumofatmostfourprimes(ifniseventhenn–3isodd,henceasumofthreeprimesq+r+s,son=3+q+r+sisasumoffourprimes).ThisisclosetotheevenGoldbachconjecture,butitseemsunlikelythatthiscanbeprovedinfullusingcurrentmethods.Sothere’sstillsomewaytogo.

PrimeNumberMysteriesMathematicshasmysteriesofitsown,andmathematicianswhotrytoresolvethemarelikedetectives.Theyseekoutclues,makelogicaldeductions,andlookforproofthattheyarecorrect.AsinSoames’scases,themostimportantstepisknowinghowtogetstarted—whatlineofthinkingmightleadtoprogress.Inmanycases,westilldon’tknow.Thismaysoundlikeanadmissionofignorance,andindeeditis.Butitisalsoastatementthatnewmathematicsstillremainstobefound,sothesubjecthasnotrundry.Theprimesarearichsourceofplausiblethingsthatwedon’tactuallyknowaretrue.Hereareafewofthem.Inallcasespndenotesthenthprime.

Agoh–GiugaConjectureAnumberpisprimeifandonlyifthenumeratorofpBp–1+1isdivisiblebyp,whereBkisthekthBernoullinumber[TakashiAgoh1990].LookthoseupontheInternetifyoureallywanttoknow:thefirstfewareB0=1,B1= ,B2= ,B3=0,B4=– ,B5=0,B6= ,B7=0,B8=– .

Equivalently:Anumberpisprimeifandonlyif

[1p–1+2p–1+3p–1+…+(p–1)p–1]+1

isdivisiblebyp[GiuseppeGiuca1950].Acounterexample,ifitexists,musthaveatleast13,800digits[David

Borwein,JonathanBorwein,PeterBorwein,andRolandGirgensohn1996].

Andrica’sConjectureIfpnisthenthprime,then

[DorinAndrica1986].ImranGhoryhasuseddataonthelargestprimegapstoconfirmthe

conjecturefornupto1.3002×1016.Thefigureplots againstnforthefirst500primes.Thenumber1isatthetopoftheverticalaxis,andallofthespikesshownarelowerthanthat.Theyseemtoshrinkasthengetsbigger,butforallweknow,theremightbeahugespikepokingoutabove1forsomeverylargen.Inorderfortheconjecturetobefalse,therewouldhavetobean

extremelylargegapbetweentwoverylargeconsecutiveprimes.Thisseemshighlyunlikely,butcan’tyetberuledout.

Plotof againstnforthefirst200primes

Artin’sConjectureonPrimitiveRootsAnyintegera,otherthan–1oraperfectsquare,isaprimitiverootmoduloinfinitelymanyprimes.Thatis,everynumberbetween1andp–1isapowerofaminusamultipleofp.Therearespecificformulasfortheproportionofsuchprimesastheirsizebecomeslarge[EmilArtin1927].

Brocard’sConjectureWhenn>1thereareatleastfourprimesbetween and [HenriBrocard1904].Thisisexpectedtobetrue;indeedmuchstrongerstatementsoughttobetrue.

Numberofprimesbetween and plottedagainstn[EricW.Weisstein,‘Brocard’sConjecture’,fromMathWorld—AWolframWebResource:http://mathworld.wolfram.com/BrocardsConjecture.html]

Cramér’sConjectureThegappn+1–pnbetweenconsecutiveprimesisnogreaterthanaconstantmultipleof(logpn)2whennbecomeslarge[HaraldCramér1936].

Cramérprovedasimilarstatementreplacing(logpn)2by ,assuming

theRiemannHypothesis—perhapsthemostimportantunsolvedprobleminmathematics,seeCabinetpage215.

Firoozbakht’sConjectureThevalueof isstrictlydecreasing[FaridehFiroozbakht1982].Thatis,

foralln.Itistrueforallprimesupto4×1018.

FirstHardy–LittlewoodConjectureLetπ2(x)denotethenumberofprimesp≤xsuchthatp+2isalsoprime.Definethetwinprimeconstant

(wherethePsymbolindicatesaproductextendingoverallprimenumbersp≥3).Thentheconjectureisthat

where~meansthattheratiotendsto1asnbecomesarbitrarilylarge[GodfreyHaroldHardyandJohnEdensorLittlewood1923].

ThereisasecondHardy–Littlewoodconjecture(below).

Gilbreath’sConjectureStartwiththeprimes

2,3,5,7,11,13,17,19,23,29,31,…

Computethedifferencesbetweenconsecutiveterms:

1,2,2,4,2,4,2,4,6,2,…

Repeatthesamecalculationforthenewsequence,ignoringsigns,andcontinue.Thefirstfivesequencesare

1,0,2,2,2,2,2,2,4,…

1,2,0,0,0,0,0,2,…

1,2,0,0,0,0,2,…

1,2,0,0,0,2,…

1,2,0,0,2,…

GilbreathandProthconjecturedthatthefirsttermineachsequenceisalways1,nomatterhowmanytimestheprocessiscontinued[NormanGilbreath1958,FrançoisProth1878].

AndrewOdlyzkoverifiedtheconjectureforthefirst3.4×1011sequencesin1993.

Goldbach’sConjectureforEvenNumbersEveryevenintegergreaterthan2canbeexpressedasthesumoftwoprimes[ChristianGoldbach1742].

T.OliveiraeSilvahasverifiedtheconjecturebycomputerforn≤1.609×1018.

Grimm’sConjectureToeachelementofasetofconsecutivecompositenumbersonecanassignadistinctprimethatdividesit[C.A.Grimm1969].

Forexample,ifthecompositenumbersare32,33,34,35,36,thenonesuchassignmentis2,11,17,5,3.

Landau’sFourthProblemIn1912EdmundLandaulistedfourbasicproblemsaboutprimes,nowknownasLandau’sproblems.ThefirstthreeareGoldbach’sconjecture(above),thetwinprimeconjecture(below),andLegendre’sconjecture(below).Thefourthis:arethereinfinitelymanyprimespsuchthatp–1isaperfectsquare?Thatis,p=x2+1forintegerx.

Thefirstfewsuchprimesare2,5,17,37,101,197,257,401,577,677,1297,1601,2917,3137,4357,5477,7057,8101,8837,12101,13457,14401,and15377.Alargerexample,bynomeansthelargest,is

p=1,524,157,875,323,883,675,049,535,156,256,668,194,500,533,455,762,536,198,787,501,905,199,875,019,052,101

x=1,234,567,890,123456789,012,345,678,901,234,567,890

In1997JohnFriedlanderandHenrykIwaniecprovedthatinfinitelymany

primesareoftheformx2+y4forintegerx,y.Thefirstfeware2,5,17,37,41,97,101,137,181,197,241,257,277,281,337,401,and457.Iwaniechasprovedthatinfinitelymanynumbersoftheformx2+1haveatmosttwoprimefactors.Close,butnobanana.

Legendre’sConjectureAdrien-MarieLegendreconjecturedthatthereisaprimebetweenn2and(n+1)2foreverypositiven.ThiswouldfollowfromAndrica’sconjecture(above)andOppermann’sconjecture(below).Cramér’sconjecture(above)impliesthatLegendre’sconjectureistrueforallsufficientlylargenumbers.Itisknowntobetrueupto1018.

Lemoine’sConjectureorLevy’sConjectureAlloddintegersgreaterthan5canberepresentedasthesumofanoddprimeandtwiceaprime[ÉmileLemoine1894,HymanLevy1963].

Theconjecturehasbeenverifiedupto109byD.Corbitt.

MersenneConjecturesIn1644MarinMersennestatedthatthenumbers2n–1areprimeforn=2,3,5,7,13,17,19,31,67,127and257,andcompositeforallotherpositiveintegersn<257.EventuallyitwasshownthatMersennehadmadefiveerrors:n=67and257givecompositenumbersandn=61,89,107giveprimes.Mersenne’sconjectureledtotheNewMersenneConjectureandtheLenstra–Pomerance–WagstaffConjecture,whichfollow.

NewMersenneConjectureorBateman-Selfridge-WagstaffConjectureForanyoddp,ifanytwoofthefollowingconditionshold,thensodoesthethird:

1p+2k 1orp=4k 3forsomenaturalnumberk.

22p–1isprime(aMersenneprime).

3(2p+1)/3isprime(aWagstaffprime).

[PaulBateman,JohnSelfridge,andSamuelWagstaffJr1989]

Lenstra–Pomerance–WagstaffConjectureThereisaninfinitenumberofMersenneprimes,andthenumberofMersenneprimeslessthanxisapproximatelyeγloglogx/log2whereγisEuler’s

constant,roughly0.577[HendrikLenstra,CarlPomerance,andWagstaff,unpublished].

Oppermann’sConjectureForanyintegern>1,thereisatleastoneprimebetweenn(n–1)andn2,andatleastanotherprimebetweenn2andn(n+1)[LudvigHenrikFerdinandOppermann1882].

Polignac’sConjectureForanypositiveevenn,thereareinfinitelymanycasesoftwoconsecutiveprimeswithdifferencen[AlphonsedePolignac1849].

Forn=2,thisisthetwinprimeconjecture(seebelow).Forn=4,itsaysthereareinfinitelymanycousinprimes(p,p+4).Forn=6,itsaysthereareinfinitelymanysexyprimes(p,p+6)withnoprimebetweenpandp+6.

Redmond–SunConjectureEveryinterval[xm,yn](thatis,thesetofnumbersrunningfromxmtoyn)containsatleastoneprime,exceptfor[23,32],[52,33],[25,62],[112,53],[37,133],[55,562],[1812,215],[433,2822],[463,3122],[224342,555][StephenRedmondandZhi-WeiSun2006].

Theconjecturehasbeenverifiedforallintervals[xm,yn]below1012.

SecondHardy–LittlewoodConjectureIfπ(x)isthenumberofprimesuptoandincludingxthen

π(x+y)≤π(x)+π(y)

forx,y≥2[GodfreyHaroldHardyandJohnLittlewood1923].Therearetechnicalreasonsforexpectingthistobefalse,butthefirst

violationislikelytooccurforverylargevaluesofx,probablygreaterthan1.5×10174butlessthan2.2×101198.

TwinPrimeConjectureThereareinfinitelymanyprimespsuchthatp+2isalsoprime.

On25December2011,PrimeGrid,a‘distributedcomputingproject’thatmakesuseofsparetimeonvolunteers’computers,announcedthelargestpairoftwinprimescurrentlyknown:

3,756,801,695,685×2666,669 1

Thesenumbershave200,700digits.Thereare808,675,888,577,436twinprimepairsbelow1018.

TheOptimalPyramidThinkofancientEgyptandyouthinkofpyramids.EspeciallytheGreatPyramidofKhufuatGiza,thelargestofthemall,flankedbytheslightlysmallerpyramidofKhafreandtherelativeminnowofMenkaure.Theremainsofover36majorEgyptianpyramidsandhundredsofsmalleronesareknown;theyrangefromhugealmostcompleteonestoholesinthegroundcontainingafewbitsofstonefromtheburialchamber—orless.

Left:TheGizapyramids:Frombacktofront:GreatPyramidofKhufu,Khafre,Menkaure,andthreeQueens’pyramids.Perspectivemakesthosebehindlooksmallerthantheyreallyare.Right:Thebent

pyramid.

Anenormousamounthasbeenwrittenabouttheshapes,sizes,andorientationsofpyramids.Mostofitisspeculative,usingnumericalrelationshipstoconstructambitiouschainsofargument.TheGreatPyramidisespeciallysusceptibletothistreatment,andithasbeenvariouslylinkedtothegoldennumber,π,andeventhespeedoflight.Therearesomanyproblemswiththiskindofreasoningthatit’shardtotakeitseriously:thedataareofteninaccurateanyway,andwithsomanymeasurementstoplaywithit’seasytocomeupwithanythingyouwant.

OneofthebestsourcesfordataisMarkLehner’sTheCompletePyramids.Amongotherthings,itliststheslopesofthefaces:theanglesbetweentheplanesformedbyatriangularfaceandthesquarebase.Afewexamples:

pyramid angle

Khufu 51°50’40”Khafre 53°10’Menkaure 51°20’25”Bent 54°27’44”(lower),43°22’(upper)

Red 43°22’Black 57°15’50”

Youcanfindmoreextensivedataat

http://en.wikipedia.org/wiki/List_of_Egyptian_pyramids

Twoobservationsspringtomind.Thefirstisthatstatingsomeoftheseanglestothenearestsecondofarc(andotherstothenearestminute)isunwise.TheblackpyramidofAmenemhatIIIatDashurhasabaseof105metresandaheightof75metres.Adifferenceof1secondofarcintheslopecorrespondstoachangeinheightof1millimetre.Admittedly,therearetracesoftheedgesofthebase,andsomefragmentsofcasingstonesmayhavesurvived,butgivenwhatremainsofthepyramid,you’dbehardputtoestimatetheoriginalslopetowithin5°ofthetruefigure.

WhatremainsoftheblackpyramidofAmenemhatIII

Theotheristhatalthoughtheslopesvarysomewhat—withinasinglemonumentinthecaseofthebentpyramid—theyhaveatendencytoclusteraround54°orso.Why?

In1979R.H.Macmillan[Pyramidsandpavements:somethoughtsfromCairo,MathematicalGazette63(December1979)251–255]startedfromthewell-attestedfactthatthepyramidbuildersusedexpensivecasingstonesontheoutsideoftheirpyramids,suchaswhiteTuralimestoneorgranite.Insidetheyusedcheapermaterials:low-gradeMokattamlimestone,mudbrick,andrubble.Soitmakessensetoreducetheamountofstonecasing.Whatshapeshouldapyramidbeifthepharaohwantsthebiggestpossiblemonumentforagivencostofcasingstones?Thatis:whichangleofslopemaximisesthevolumeforafixedareaofthefourtriangularfaces?

Left:Slicingapyramid.Right:Maximisingtheareaofanisoscelestriangle,orequivalentlyarhombuswithgivenside.

Thisisaniceexerciseincalculus,butitcanalsobesolvedgeometricallyusingaclevertrick.Slicethepyramidinhalfverticallythroughadiagonalofthebase(shadedtriangle).Thisyieldsanisoscelestriangle.Thevolumeofthehalf-pyramidisproportionaltotheareaofthistriangle,andtheareasoftheslopingfacesofthehalf-pyramidareproportionaltothelengthsofthecorrespondingsidesofthetriangle.Sotheproblemisequivalenttofindingtheisoscelestriangleofmaximalareawhenthelengthsofthetwoequalsidesarefixed.

Reflectingthetriangleinitsbase,thisisequivalenttofindingtherhombusofmaximalareawithagivenlengthofside.Theanswerisasquare(orientedinadiamondposition).Sotheanglesofeachtriangularsectionofthiskindare90°atthetop,and45°forthetwobaseangles.Itfollowsbybasictrigonometrythattheangleofslopeofafaceofthepyramidis

whichisclosetotheaveragefigureforactualpyramids.Macmillanmakesnoclaimsaboutwhatthistellsusaboutbuildingpyramids;

hismainpointisthatit’saneatexerciseingeometry.However,theMoscowmathematicalpapyruscontainsaruleforfindingthevolumeofatruncatedpyramid(onewiththetopcutoff),andaproblemshowingthattheEgyptiansunderstoodsimilarity.Italsoexplainshowtofindtheheightofapyramidfromitsbaseandslope.Moreover,boththispapyrusandtheRhindmathematicalpapyrusexplainhowtofindtheareaofatriangle.SotheEgyptianmathematicianscouldhavesolvedMacmillan’sproblem.

MoscowmathematicalpapyrusProblem14:findingthevolumeofatruncatedpyramid

Intheabsenceofapapyruscontainingthatexactcalculation,there’snoconvincingreasontosupposethattheydid.Wedon’thaveanyevidencethattheywereinterestedinoptimisingtheshapeoftheirpyramids.Eveniftheywere,theycouldhaveestimatedtheshapeusingclaymodels.Ormadeaneducatedguess.Ortheshapecouldhaveevolvedtowardstheleastexpensiveone:buildersandpharaohsarelikethat.Alternatively,theangleofslopemighthavebeendeterminedbyengineeringconsiderations:thebentpyramidiswidelybelievedtobetheshapeitisbecausehalfwayupitstartedtocollapse,sotheslopewasreduced.Thatsaid,thislittlepieceofpyramidmathematicshasmoregoingforitthanconnectionstothespeedoflight.

TheSignofOne:PartTwoFromtheMemoirsofDrWatsup

Soamesbeganstridingupanddowntheroomlikeamanpossessed.InwardlyIgavealoud“huzzah!”—forIcouldseethathewashooked.NowIwouldreelhiminoutofthedarkdepressionintowhichhehadfallen,andridmyselfofBolivianfuneraldirgestoboot.

“Wemustbemoresystematic,Watsup!”hedeclared.“Inwhatway,Soames?”“Inamoresystematicway,Watsup.”Theensuingsilenceledhimtobeless

obscure.“Wemustlistsmallnumbersthatcanbederivedfromjusttwo1’s.Byputtingthosetogetherwecan—well,you’llseeinaminute,I’msure.”

SoSoameswrotedown:

atwhichpointSoamesgroundtoahalt.“Iadmitthat7and8constitutetemporarylacunae,”hesaid.“Nevermind,

allowmetocontinue:

9=1/.10=1/.111=11”

“IconfessIdonotyet—”“Beassured,Watsup,youwill.Suppose,forargument’ssake,thatwecan

express7and8usingtwo1’s.Thenwewouldhavecontrolofallnumbersfrom0to11.So,givenanynumbernexpressibleusingtwo1’s,wewouldbeableto

expresseverythingbetweenn–11andn+11usingfour1’s—merelybysubtractingoraddingtheexpressionsonmysystematiclist.”

“Ah,Iseenow,”Isaid.“Youusuallydo,onceI’vetoldyou,”herepliedacerbically.“ThenletmecontributeanewthoughttoshowthatIhaveunderstood!Since

weknowhowtoexpress24usingtwo1’s,forexampleas !,weimmediatelyexpresseverynumberfrom24–11to24+11usingfour1’s.Thatis,therangefrom13to35,inclusive.”

“Exactly!Ithinkweneednotwritethoseexpressionsdown.”“No,Aha!Wecangetfurther!Look:

“Yes,”hereplied.“Butbeforeyourexcitementcarriesyouintorealmsuntold,Iwillremindyouthatwedonotyethaveexpressionsfor7and8usingonlytwo1’s.”

Ilookedsuitablycrestfallen.Butthenawildthoughtstruckme.“Soames?”Iaskedtentatively.

“Yes?”“Factorialsmakenumbersbigger?”Henodded,irritably.“Andsquarerootsmakethemsmaller?”“Agreed.Gettothepoint,man!”“Andfloorsandceilingsroundthingsofftowholenumbers?”Icouldseetherealisationdawningonhisface.“Stoutfellow,Watsup!Yes,I

seeitnow.Weknow,forexample,howtoexpress24usingtwo1’s.Thereforewecanalsoexpress24!usingtwo1’s,andthatis”—hiseyebrowsnarrowed—“620,448,401,733,239,439,360,000.Whosesquarerootis”—hisfacereddenedashecarriedoutthementalarithmetic—“887,516.46,whosesquarerootis942.08,whosesquarerootis30.69.”

“Sowecanexpress30and31usingonlytwo1’s,”Isaid.“Namely:

Noneofwhichhelpsusexpress7and8withtwo1’s,ofcourse,butifwecoulddothat,wecouldextendtherangeofnumbersto31+11,whichis42.Allofwhichargues,asyousocogentlyputit,Soames,thatweshouldbesystematic.I

proposethatwenowinvestigaterepeatedsquarerootsoffactorialsofnumbersthatwecanexpresswithtwo1’s.”

“Agreed!Anditisimmediatelyobvious,”saidSoames,“thatsuchanexpressionfor7immediatelyyieldsonefor8.”

“Uh—isit?”“Naturally.Since7!=5040,whosesquarerootis70.99,whosesquarerootis

8.42,wededucethat

So,notforthefirsttimeinhumanhistory,thekeytothemysteryisthenumber7!”Bywhich,dearreader,hewasplacinganemphasisonthenumber7,notreferringtoitsfactorial.Pleasepayattention,Ihaveexplainedthisbefore.

Soames’sbrowfurrowed.“Icandoitusingadoublefactorial.”“Youmeanafactorialofafactorial?”“No.”“Asubfactorial?You’venotyetexplained—”“No.Thedoublefactorialisatrifleobscure;itis

n!!=n×(n–2)×(n–4)×…×4×2

whenniseven,and

n!!=n×(n–2)×(n–4)×…×3×1

whennisodd.So,forexample,

6!!=6×4×2=48

whosesquarerootis6.92,whoseceilingis7.”Imeeklywrotedown

ButSoamesremaineddissatisfied.“Theproblem,Watsup,isthatbyintroducingevermoreobscurearithmetical

functions,wecouldexpressanynumberwhatsoeverwithease.WemightusePeano,forinstance.”

Iobjectedvociferously.“Soames,youknowthatourlandladycomplainsincessantlyaboutyourclarinet.Shewouldneverpermitapiano!”

“GiuseppePeanowasanItalianlogician,Watsup.”“Tobehonest,thatmightnotmakemuchdifference.I’mnotsurethatMrs

Soapsudswould—”“Quiet!InPeano’saxiomatisiationofarithmetic,thesuccessorofanyinteger

nis

s(n)=n+1

SoPeanocouldwrite

1=12=s(1)3=s(s(1))4=s(s(s(1)))5=s(s(s(s(1))))

andthepatterncontinuesindefinitely.Everyintegerwouldbeexpressibleusingjustone1.Or,forthatmatter,justone0,since1=s(0).Itistootrivial,Watsup.”

Canyoufindawaytowrite7usingonlytwo1’sandnotusinganythingmoreesotericthanthefunctionsSoamesandWatsupemployedbeforetheystartedarguingaboutdoublefactorialsandsuccessors?Seepage279fortheanswer.

SoamesandWatsuphavenotfinishedyet.TheSignofOnecontinuesonpage115.

InitialConfusion

R.H.Bing

R.H.BingwasanAmericanmathematician,borninTexas,whospecialisedinwhatbecameknownasTexantopology.WhatdoestheR.H.standfor?Well,hisfatherwasRupertHenry,buthismotherfeltthatthissoundedtooBritishforTexas,sowhenhewaschristenedshecutitdowntothebareinitials.SoR.H.standsforR.H.,nothingmore.Thiscausedacertainamountofpuzzlement,butnothingtooserious,untilBingappliedforavisatovisitsomewhereorother.Whenaskedhisname,anticipatingtheusualreaction,hesaiditwas“R-onlyH-onlyBing”.

HereceivedavisamadeouttoRonlyHonlyBing.

Euclid’sDoodleThisisamathematicalmysterythatwassolvedovertwothousandyearsago,andusedtobetaughtinschools,butnotanymore—forsensiblereasons.However,it’sworthknowing,becauseit’sfarmoreefficientthanthemethodthat’susuallytaughtinstead.Anditlinksuptoallsortsofimportantbitsofmathematicsathigherlevels.

Peopleliketodoodle.Youseethemonthephone,engagedinconversation,idlyfillinginalltheo’sonapageofthenewspaperwithaballpointpen.Ordrawingwigglylinesthatgoroundandroundlikeirregularspirals.Thissenseoftheword‘doodle’,whichoriginallymeantafool,seemstohavebeenintroducedbythescreenwriterRobertRiskininthe1936comedymovieMrDeedsGoestoTown;MrDeedsrefersto‘doodle’asscribblesthatcanhelppeoplethink.

Ifamathematiciandoodled—andmostdo—theymightwellgetroundtodrawingarectangle.Whatcanyoudotoarectangle?Youcanfillitin,youcandrawspiral-likecurvesroundtheedge…oryoucancutoffasquarefromoneendtomakeasmallerrectangle.Thenit’sonlynatural,andtypicalofthedoodlingmentality,todothesameagain.

Whathappens?Youmightcaretotryafewrectanglesbeforereadingon.OK,herewego.Istartedwithalong,thinrectangle,andhere’swhat

happened.

Mydoodle

EventuallyIgottoasmallsquare,andranoutofrectangle.Doesthisalwayshappen?Doeseveryrectangleeventuallygetgobbledup?

Nowthat’sagoodquestionforamathematiciantothinkabout.Whatsizewasmyrectangle?Well,thefinalpictureshowsthat:

Thecombinedsidesoftwolittlesquaresequalthesideofamediumsquare.Thecombinedsidesoftwomediumsquaresandonelittleonemakethesideofabigone,andalsogiveonesideoftherectangle.Thecombinedsidesofthreebigsquaresandamediumonemaketheothersideoftherectangle.

Ifthesmallsquarehasside1unit,thenthemediumonehasside2andthelargeonehasside2×2+1=5.Theshortsideoftherectangleis5,andthelongsideis3×5+1=17.SoIstartedwitha17×5rectangle.

That’sinteresting:bylookingatthewaythesquaresfittogether,Icanworkouttheshapeofmyrectangle.Amoresubtleimplicationis:iftheprocessstops,thesidesoftheoriginalrectanglearebothintegermultiplesofthesamething:thesideofthelastsquareremoved.Inotherwords,theratioofthetwosidesisoftheformp/qforintegerspandq.Whichmakesitarationalnumber.

Thisideaiscompletelygeneral:ifthedoodlestops,theratioofthesidesoftherectangleisarationalnumber.Infact,theconverseisalsotrue:iftheratioofthesidesoftherectangleisarationalnumber,thedoodlestops.Sodoodlesthatstopcorrespondpreciselyto‘rationalrectangles’.

Toseewhy,let’slookmorecloselyatthenumbers.Thepicturesineffecttellusthis:

17–5=1212–5=77–5=2

Nowwehavea5×2rectangleleftandwehavetomovetothemediumsquare

5–2=33–2=1

Nowwehavea2×1rectangleleftandwehavetomovetoasmallsquare

2–1=11–1=0

Stop!Anditmuststop,becausetheintegersinvolvedarepositive,andtheyaregettingsmallerateachstage.Theymustdo,we’resubtractingfromthemorleavingthemalone.Now,asequenceofpositiveintegerscan’tdecreaseforever.Forexample,ifyoustartwithamillionandthendecrease,youhavetostopafteratmostamillionsubtractions.

Morecompactly,thedoodletellsusthat

17dividedby5gives3withremainder25dividedby2gives2withremainder12dividedby1goesexactlywithzeroremainder

andtheprocessstopsoncetheremainderiszero.Euclidusedhisdoodletosolveaprobleminarithmetic:giventwointegers,

calculatetheirhighestcommonfactor.Thisisthelargestintegerthatdividesbothexactly;itisusuallyabbreviatedtohcf.Anotherphraseisgreatestcommondivisor,gcd.Forinstance,ifthenumbersare4,500and840,thehcfis120.

ThewayIwastaughttodothisatschoolistofactorisethetwonumbersintoprimes,andseewhatfactorstheyhaveincommon.Forinstance,supposewewantthehcfof68and20.Factoriseintoprimes:

68=22×1720=22×5

Thehcfis22=4.Thismethodislimitedtonumbersthataresmallenoughtobefactorised

quicklyintoprimes.It’shopelesslyinefficientforlargernumbers.TheancientGreeksknewamoreefficientmethod,aprocedurethattheygavethefancynameanthyphairesis.Inthiscaseitgoeslikethis:

68dividedby20gives3withremainder820dividedby8gives2withremainder48dividedby4goesexactlywithzeroremainderStop!

Thisisthesameasthepreviouscalculationwith17and5,butnowallthenumbersarefourtimesthesize(exceptforhowmanytimeseachdividesintotheother,whichstaysthesame).Ifyoudothedoodlewitha68×20rectangleyougetthesamepicturesasbefore,butthefinalsmallsquareis4×4,not1×1.

ThetechnicalnameisEuclid’salgorithm.Analgorithmisarecipeforacalculation.EuclidputthisoneinhisElements,andheuseditasthebasisofhistheoryofprimenumbers.Insymbols,thedoodlegoeslikethis.Taketwopositiveintegersm≤n.Startwiththepair(m,n)andreplaceitby(m,n–m)innumericalorder,smallestfirst:thatis,transform

(m,n) (min(m,n–m),max(m,n–m))

whereminandmaxaretheminimumandmaximumrespectively.Repeat.At

eachstagethelargestnumberinthepairgetssmaller,soeventuallytheprocessstopswithapair(0,h),say.Thenhistherequiredhcf.Theproofiseasy:anyfactorofbothmandnisalsoafactorofbothmandn–m,andconversely.Soateachstepthehcfremainsthesame.

Thismethodisgenuinelyefficient:youcanuseitbyhandforreallybignumbers.Toproveit,here’saquestionforyou.Findthehcfof44,758,272,401and13,164,197,765.

Seepage279fortheanswer.

EuclideanEfficiencyHowefficientisEuclid’salgorithm?

Choppingoffonesquareatatimeissimplerfortheoreticalpurposes,butthemorecompactform,intermsofdivisionwithremainder,isthebestonetouseinpractice.Thistelescopesallcutsusingafixedsizeofsquareintoasingleoperation.

Mostofthecomputationaleffortoccursinthedivisionstep,sowecanestimatetheefficiencyofthealgorithmbycountinghowmanytimesthisstepgetsused.ThefirstpersontoinvestigatethisquestionwasA.A.-L.Reynaud,andin1811heprovedthatthenumberofdivisionstepsisatmostm,thesmallerofthetwonumbers.Thisisaverypoorestimate,andhelatergotitdowntom/2+2,notmuchbetter.In1841P.-J.-E.Finckreducedtheestimateto2log2m+1,whichisproportionaltothenumberofdecimaldigitsinm.In1844GabrielLaméprovedthatthenumberofdivisionstepsisatmostfivetimesthenumberofdecimaldigitsofm.Soevenfortwonumberswith100digits,thealgorithmgetstheanswerinnomorethan500steps.Ingeneral,youcan’tdoitthatquicklyusingprimefactors.

What’stheworst-casescenario?LaméprovedthatthealgorithmrunsmostslowlywhenmandnareconsecutivemembersoftheFibonaccisequence

1123581321345589…

inwhicheachnumberisthesumoftheprevioustwo.Forthesenumbers,exactlyonesquaregetschoppedoffeachtime.Forinstance,withm=34,n=55,weget

55dividedby34gives1withremainder2134dividedby21gives1withremainder1321dividedby13gives1withremainder813dividedby8gives1withremainder58dividedby5gives1withremainder35dividedby3gives1withremainder23dividedby2gives1withremainder12dividedby1goesexactly.

It’sanunusuallylongcalculationforsuchsmallnumbers.Mathematicianshavealsoanalysedtheaveragenumberofdivisionsteps.

Withfixedn,thenumberofdivisionstepsaveragedoverallsmallermisapproximately

whereC,calledPorter’sconstant,is

Here ’(2)isthederivativeofRiemann’szetafunctionevaluatedat2,andγisEuler’sconstant,about0.577.Itwouldbehardtofindasensibleproblemleadingtoamorecomprehensiveselectionofmathematicalconstantsinoneformula.Theratioofthisformulatotheexactanswertendsto1asngetslarger.

123456789TimesXSometimesverysimpleideasleadtomysteriousresults.Trymultiplying123456789by1,2,3,4,5,6,7,8,and9.Whatdoyounotice?Whendoesitgowrong?

Seepage279fortheanswers.Foranextension,Seepage145.

TheSignofOne:PartThreeFromtheMemoirsofDrWatsup

HeapsofpapercoveredinarcanescribblesweresproutinglikemushroomsfromeveryflatsurfaceinSoames’slodgings.This,youunderstand,wasnotunusual;MrsSoapsudsoftenberatedhimabouthisdeeplitterfilingsystem,tonoavail.Butonthisoccasionthescribblesweresums.

“Icanobtain8usingtwo1’s,withoutinvolvingahypotheticalexpressionfor7,”Iannounced.“Infact,

ButforthelifeofmeIcannotderive7.”“Thatonedoesseemtricky,”Soamesagreed.“Butyourresultleadsto

progressinotherways:

whereofcoursewesubstituteyourexpressionfor8wherenecessary.Icouldwriteitoutinfull—”

“No,no,Soames,Iamalreadyconvinced!”“Butnowwehavetwomorelacunaeat12and13.However,Watsup,I

suspecttheseproblemsarerelated.Letmesee…Well,

andwealreadyhave15usingonlytwo1’s.Then

andfurther

andfinally

whichresolvestheissueentirelysatisfactorily.Sosubstitutinginturnforthevariousnumbers,wefindthat

Iammortifiednottohaveseenthatimmediately.”“Isthatthesimplestsolution,Soames?”saidI,gulping.“Ihopenot!”“Ihavenoidea.Perhapssomeingeniouspersoncoulddobetter.Itishardto

besurewiththesematters.Iamsuretheywillinformusbytelegramiftheycanbetterourownfeebleefforts.”

“Anway,”Isaid,“ifwecanexpressanyintegernwithtwo1’s,wecannowexpresstheentirerangefromn–17ton+17.”

“Exactly,Watsup.Ourquestbecomessimplerbythemoment.Allweneedisaseriesofnumbers,eachexceedingthepreviousonebynomorethan35,sothattheirrangesabutoroverlap.Thatwillenableustoreachthelargestsuchnumberplus17.”

“Whichmeans—”Ibegan—“Thatweshouldbesystematic!”“Quite.”“Wehadalreadyreached…remindme,Watsup.Consultthoseextensive

notesofyours.”Idelvedintoseveralpilesofdocuments,eventuallyfindingmynotebook

beneathastuffedskunk.“Wehadreached32,Soames,ifweincludeyourincidentalremarkearlierwhenseeking7.”

“And,ofcourse,

saidhe.“Verygood.Soideallyweneedtoexpress78,103,138,andsoon,withtwo1’s.Butwecanusesmallernumbersifthosearemoreconvenient.Providedthattheincreaseisatmost35fromonetothenext.”

Severalhoursofintensecomputation,andmoreheapsofpaper,ledtoashortbutvitallist:

butlittlemore.“PerhapsIwastoohastyindismissingdoublefactorials,Watsup.”

“Verylikely,Soames.”Soamesnoddedandwrote

105=7!!

Then,inasuddenfitofinspiration,headded

exclaiming“Ifwecanfindawaytowrite18withtwo1’s,thenweextendtherangesurroundinganyintegernexpressibleusingtwo1’s:wecanthendealwithn–20ton+20.”Hepausedforbreath,andadded“Failingthat,theonlygapswouldben–18andn+18,whichperhapswecouldderiveinotherways.”

“Ithinkitistimewetookstock,”Isaid.Iperusedourscribblednotes.“Itseemstomethatwehadexpressedallnumbersfrom1to33usingfour1’s.Then

needonlytwo1’s,soweimmediatelyfillineverythingbetween26and61.Thereisagapat62(becausethatis44+18andwearestuckon18ifonlytwo1’sareallowed)butwecando63and64.Now,basedon80,wecancontinueto97.Thenwearestuckon98,but99and100canbeachieved.”

“Moreeasily,inpointoffact,”saidSoames:

“Sowearecompleteupto100,”Isaid,“withtheexceptionsof62and98.”“But98istakencareofby105,alongwithallnumbersupto122,”said

Soames.“Oh,Ihadforgottenwecoulddo105withtwo1’s.”“Andsince120=5!,alsoexpressibleusingtwo1’s,wecanreach137.

Indeed,wealsoget139and140.”“Sotheonlygapsupto140are62and138,”Isaid.“Soitseems,”saidSoames.“Iwonderifthosegapscanbefilledbysome

othermethod?”Canyoufindawaytowrite62and138withfour1’s,notusinganything

moreesotericthanthefunctionsSoamesandWatsuphaveusedsofar?Seepage280fortheanswer.

SoamesandWatsupstillhaven’tfinished.Buttheendisnigh:TheSignofOneconcludesonpage126.

TaxicabNumbers

SrinivasaRamanujan

SrinivasaRamanujanwasaself-taughtIndianmathematicianwithanamazingtalentforformulas—usuallyverystrangeones,yethavingtheirownkindofbeauty.HewasbroughttoEnglandbyCambridgemathematiciansGodfreyHaroldHardyandJohnEdensorLittlewoodin1914.By1919hewasterminallyillwithlungdisease,andhediedinIndiain1920.Hardywrote:

IrememberoncegoingtoseehimwhenhewaslyingillatPutney.Ihadriddenintaxi-cabNo.1729,andremarkedthatthenumberseemedtoberatheradullone,andthatIhopeditwasnotanunfavourableomen.“No,”hereplied,“itisaveryinterestingnumber;itisthesmallestnumberexpressibleasthesumoftwo[positive]cubesintwodifferentways.”

Theobservationthat

1729=13+123=93+103

wasfirstpublishedbyBernardFrénicledeBessyin1657.Ifnegativecubesarepermittedthenthesmallestsuchnumberis

91=63+(–5)3=43+33

Numbertheoristshavegeneralisedtheconcept.Thenthtaxicabnumber,Ta(n),isthesmallestnumberthatcanbeexpressedasasumoftwopositivecubesinnormoredistinctways.

In1979HardyandE.M.Wrightprovedthatsomenumberscanbeexpressed

asthesumofanarbitrarilylargenumberofpositivecubes,soTa(n)existsforalln.However,todateonlythefirstsixareknown:

Ta(1)=2=13+13Ta(2)=1729=13+123=93+103Ta(3)=87539319=1673+4363=2283+4233

=2553+4143Ta(4)=6963472309248

=24213+190833=54363+189483=102003+180723=133223+1663083

Ta(5)=48988659276962496=387873+3657573=1078393+3627533=2052923+3429523=2214243+3365883=2315183+3319543

Ta(6)=24153319581254312065344=5821623+289062063=30641733+288948033=85192813+286574873=162180683+270932083=174924963+265904523=182899223+262243663

Ta(3)wasdiscoveredbyJohnLeechin1957.Ta(4)wasfoundbyE.Rosenstiel,J.A.DardisandC.R.Rosenstielin1991.Ta(5)wasfoundbyJ.A.Dardisin1994andconfirmedbyDavidWilsonin1999.In2003C.S.Calude,E.Calude,andM.J.DinneenestablishedthatthenumberstatedaboveisprobablyTa(6),andin2008UweHollerbachannouncedaproof.

TheWaveofTranslation

JohnScottRussell

Mathematicalresearchonhorseback?Whynot?Inspirationstrikeswheneveritdoes.Youdon’tgettochoose.In1834,JohnScottRussell,aScottishcivilengineerandnavalarchitect,was

ridinghishorsealongsideacanal,whenhenoticedsomethingremarkable:

Iwasobservingthemotionofaboatwhichwasrapidlydrawnalonganarrowchannelbyapairofhorses,whentheboatsuddenlystopped—notsothemassofwaterinthechannelwhichithadputinmotion;itaccumulatedroundtheprowofthevesselinastateofviolentagitation,thensuddenlyleavingitbehind,rolledforwardwithgreatvelocity,assumingtheformofalargesolitaryelevation,arounded,smoothandwell-definedheapofwater,whichcontinueditscoursealongthechannelapparentlywithoutchangeofformordiminutionofspeed.Ifolloweditonhorseback,andovertookitstillrollingonatarateofsomeeightorninemilesanhour,preservingitsoriginalfiguresomethirtyfeetlongandafoottoafootandahalfinheight.Itsheightgraduallydiminished,andafterachaseofoneortwomilesIlostitinthewindingsofthechannel.Such,inthemonthofAugust1834,wasmyfirstchanceinterviewwiththatsingularandbeautiful

phenomenonwhichIhavecalledtheWaveofTranslation.

Hewasintriguedbythisdiscovery,becausenormallyindividualwavesspreadoutastheytravel,orbreaklikesurfonabeach.Heconstructedawavetankathishome,andcarriedoutaseriesofexperiments.Theserevealedthatthiskindofwaveisverystable,anditcantravelalongwaywithoutchangingshape.Wavesofdifferentsizestravelatdifferentspeeds.Ifonesuchwavecatchesupwithanotherone,itemergesinfrontafteramorecomplicatedinteraction.Andabigwaveinshallowwaterwilldivideintotwo:amediumoneandasmallone.

Thesediscoveriespuzzledthephysicistsofthetime,becausetheircurrentunderstandingoffluidflowcouldnotexplainit.Infact,GeorgeAiry,adistinguishedastronomer,andGeorgeStokes,theleadingauthorityonfluiddynamics,hadtroublebelievingit.WenowknowthatRussellwasright.Inappropriatecircumstances,nonlineareffects,beyondthescopeofthemathematicsofhisera,counteractthetendencyofawavetospreadoutbecausethespeedofthewavedependsonitsfrequency.Theseeffectswerefirstunderstoodaround1870byLordRayleighandJosephBoussinesq.

In1895DiederikKortewegandGustavdeVriescameupwiththeKorteweg–deVriesequation,whichincludedsucheffects,andshowedthatithassolitarywavesolutions.Similarresultswerederivedforotherequationsofmathematicalphysics,andthephenomenonacquiredanewname:solitons.AseriesofmajordiscoveriesledPeterLaxtoformulateverygeneralconditionsforequationstohavesolitonsolutions,andexplainedthe‘tunnelling’effect.Itismathematicallyquitedifferentfromthewaythatshallowwavesinteractbysuperposingtheirshapes,liketwosetsofripplescrossingeachotheronapond,whichisadirectconsequenceofthemathematicalformofthewaveequation.Soliton-likebehaviouroccursinmanyareasofscience,fromDNAtofibreoptics.Thishasledtoawiderangeofnewphenomenawithnameslikebreathers,kinks,andoscillons.

There’salsoatantalisingideathathasnotyetbeenmadetowork.Fundamentalparticlesinquantummechanicssomehowcombinetwoapparentlydifferentcharacteristics.Likemostthingsataquantumlevel,theyarewaves,yettheyhangtogetherinaparticle-likeclump.Physicistshavetriedtofindequationsthatrespectthestructureofquantummechanicsbutallowsolitonstoexist.Theclosesttheyhavecomesofarisanequationthatproducesaninstanton:thiscanbeinterpretedasaparticlethathasaveryshortlifetime,winkingintoexistencefromnowhereanddisappearingimmediately.

RiddleoftheSands

Barchandunes.Left:ParacasNationalPark,Peru.Right:Hellespontusregion,fromMarsReconnaissanceOrbiter.

Sanddunesformavarietyofpatterns:linear,transverse,parabolic,…Oneofthemostintriguingisthebarchan,orcrescent,dune.ThenamecomesfromTurkestan,andissaidtohavebeenintroducedintogeologyin1881bytheRussiannaturalistAlexandervonMiddendorf.BarchanscanbefoundinEgypt,Namibia,Peru,…andevenonMars.Theyarecrescent-shaped,comeinarangeofsizes,andtheymove.Theyformswarms,interactwitheachother,breakupandjointogether.Inrecentyearsmathematicalmodellinghasprovidedalotofinsightintotheirshapesandbehaviour,butmanymysteriesremain.

Dunesareformedbytheinteractionofwindandsandgrains.Theroundedendofabarchanfacesintotheprevailingwind,whichpushesthesandupthefrontoftheduneandroundthesides,whereitformstwotrailingarmsthatgiveititscharacteristiccrescentshape.Atthetopofthedune,thesandtumblesoverandissuckeddownthe‘slipface’betweenthearms.Alargevortexofrotatingaircalledaseparationbubblescoursoutthespacebetweenthearms.

Left:Schematicofabarchanandtheseparationbubble.Right:SimulationbyBarbaraHorvatofthemotionofsandgrains,computedfromamathematicalmodel.

Barchansbehavelikesolitons(seepreviousitem),althoughtechnicallythey

differinsomerespects.Asthewindblowsthemalong,thesmalldunestravelfasterthanthebigones.Ifasmalldunecatchesupwithabigone,itappearstobeabsorbed,butafteratimethebigbarchanspitsoutasmallone,almostasthroughthesmallonehadtunnelledthroughthebigone.Thesmallonethenheadsoff,fasterthanthelumberingbeastbehindit.

Intheirpaper,VeitSchwa¨mmleandHansHerrmanncommentonthesimilaritiesanddifferencesbetweenbarchancollisionsandsolitons.Thefigureshowswhathappensifthetwodunesareofsimilarsizes.Initially(a)thesmallerduneisbehindthelargeronebutmovingfaster.Ithitsthebackofthelargerone(b)andclimbsupitswindwardface,butgetsstuckpartway(c).Thenthefrontseparatestoformasmallerdune(d).

SimulatedcollisionofasmallbarchanandalargeronebyVeitSchwa¨mmleandHansHerrmann.(a)Time0:smalldunebehindbigone.(b)After0.48years:smalldunecatchesbigoneandtheycollide.(c)After

0.63years:dunesmixedtogether.(d)After1.42years:smallduneinfrontofbigone.

Forsomecombinationsofheights,theemergingduneisbiggerthanthesmalleronewasoriginally,whereasforothersitissmaller.Thisbehaviourdiffersfromthatofsolitons,wherebothwavesendupthesamesizeastheybegan.However,thereisanintermediaterangeofheightcombinationsforwhichthedunesretaintheirsizesandvolumesexactly.Inthesecasestheybehavelikesolitons.

Ifthesmallduneisalotsmallerthanthebigone,itjustgetsgobbledupandasingle,larger,barchanforms.Ifthedifferenceinheightsismoderate,thecollisioncanresultin‘breeding’:twosmallbarchansappearatthetipsofthehornsofthelargerone,andheadoffinfrontofit.Realbarchansdoallthesethings.Thedynamicsofbarchandunesisricherthanthatofconventionalsolitons.

EskimoπWhyisπonly3intheArctic?

Everythingshrinksinthecold.

TheSignofOne:PartFour—ConcludedFromtheMemoirsofDrWatsup

“Well,thisisaprettypickle,”Imuttered.“Agherkin,Ibelieve,”saidSoames,pluckingthevinegar-soakedvegetable

fromthejarandconsumingitwithrelish.Iputtherelishbackinthepantry,alongwiththepicklejar.“Wedohavetheoption,”Soamesremarked,“ofmultiplyingnumbersby3,

9,or10usingonlyoneextra1.Wemerelydivideby ,.i,or.”“ThenIhaveit!”Icried:

62=63–1=7×9–1=7/. –1

recallingthatwecanexpress7usingonlytwo1’s—indeed,inatleasttwodifferentways.”

“Leavingonly138toperplexus.”“Itis3×46,”Imused.“Canweachieve46usingjustthree1’s?Thenwe

coulddividethatby asyousuggested.”Asystematicsearchoffloorsandceilingsofrepeatedsquarerootsof

factorialsledustoanunexpecteddiscovery:itispossibletoachieve46withonlytwo1’s.Ipresentonlythesolution:theroutetoitsdiscoveryinvolvedmanydeadendsandfailures.Startwitharepresentationof7usingtwo1’s,forexample,

Thenobservethat

Workingbackwardsandsubstitutingtheformulasforthenumbers,theendresultexpresses138withjustthree1’s.

“ShallIwriteitoutexplicitly,Soames?”“Goodheavensno!Anyonewhowishestoseethefullformulacandothat

forthemselves.”Buoyedbythisunanticipatedsuccess,Iwantedtocontinueevenfurtherwith

ourlist.ButSoamesmerelyshrugged.“Perhapstheproblemmeritsfurthercomputation.Perhapsnot.”

Athoughtstruckme.“Mightweprovethateverynumbercanbeobtainedwithfour1’s—perhapsevenfewer—byiteratingfloorsandceilingsofrepeatedsquarerootsoffactorials?”

“Itisaplausibleconjecture,Watsup,buttobefrankIseenoroutetoaproof,andthestrainofsomuchmentalarithmeticisbeginningtotell.”

Hewassinkingbackintodepression.DesperatelyIsuggested“Youcouldtrylogarithms,Soames.”

“Ithoughtofthoserightatthestart,Watsup.Youmaybesurprisedtohearthatusingnothingmorethanlogarithmstheexponentialfunction,andtheceilingfunction,anypositiveintegercanbeexpressedusingonlyone1.”

“No,no,Imeantusinglogarithmsasacomputationalaid,notintheformula—”ButSoamesignoredmyprotestations.

“Recallthattheexponentialfunctionis

exp(x)=exwheree=2.71828…

anditsinversefunctionisthenaturallogarithm

log(x)=whicheverysatisfiesexp(y)=x

Isthatnotso,Watsup?”Iaffirmedthattothebestofmyknowledgeitwas.“Thenwemerelyobservethat

n+1=[log([exp(n)])]

whoseproofisstraightforward.”Igawpedathim,butmanagedtogetoutastrangled“Ofcourse,Soames.”“Thenweiterate:

1=12=[log([exp(1)])]3=[log([exp([log([exp(1)])])])]

4=[log([exp([log([exp([log([exp(1)])])])])])]

and—”Ihastilygraspedhiswritinghand.“Yes,Soames,Iunderstand.Itisathinly

disguisedvariationonthePeanomethod,whichwerejectedearlierbecauseofitstriviality.”

“Sothegameisnolongerafoot,Watsup,ifexponentialsandlogarithmsarepermitted.”

Iconcurred,withsomesadness,forimmediatelyhetookuphisclarinetandcommencedtoplayarhythmlessatonalcompositionbysomeobscureEasternEuropeancomposer.Itsoundedlikeacatcaughtinamangle.Atone-deafcat.Withasorethroat.

Hisblackmoodwouldnowbeunshakeable.HereEndsTheSignofOne.ExceptthatIstillhaven’ttoldyouwhatasubfactorialis.Thatcomesnext.

SeriouslyDerangedTimetoexplainsubfactorials.

Supposethatnpeopleeachownahat.Theyallpickuponehatandputiton.Inhowmanydifferentwayscanthisbedonesothatnoonegetstheirownhat?Suchanassignmentiscalledaderangement.

Forexample,iftherearethreepeople—Alexandra,Bethany,andCharlotte,say—thentheirhatscanbeassignedinsixways:

ABCACBBACBCACABCBA

ForABCandACB,Alexandragetsherownhat,sothesearenotderangements.ForBAC,Charlottegetsherownhat.ForCBA,Bethanygetsherownhat.Thatleavestwoderangements:BCAandCAB.

Withfourpeople—supposeDeirdrejoinsthegroup—thereare24arrangements:

but15ofthem(crossedout)assignsomeonetheirownhat.(RemoveanythingwithAinthefirstposition,Binthesecond,Cinthethird,orDinthefourth.)Sothereare9derangements.

Thenumberofderangementsofnobjectsisthesubfactorial(denotedby!norn¡).Thishasanumberofdefinitions.Thesimplestisprobably

Itsvaluesbegin

!1=0!2=1!3=2!4=9!5=44

!6=265!7=1,854!8=14,833!9=133,496!10=1,334,961

TossingaFairCoinIsn’tFair

Tossingacoin

Thefaircoinisastapleofprobabilitytheory,equallylikelytolandheadsortails.Itisgenerallyconsideredtobetheepitomeofrandomness.Ontheotherhand,acoincanbemodelledasasimplemechanicalsystem,andassuchitsmotioniscompletelydeterminedbytheinitialconditionswhenitistossed—mainlytheverticalvelocity,theinitialrateatwhichitspins,andtheaxisofspin.Thismakesthemotionnon-random.Sowheredoestherandomnessinacointosscomefrom?I’llcomebacktothatafterdescribingarelateddiscovery.

PersiDiaconis,SusanHolmes,andRichardMontgomeryhaveshownthattossinga‘fair’coinisn’tactuallyfair.Thereisasmallbutdefinitebias:whenacoinistossed,itisslightlymorelikelytolandthesamewayupasitsinitialorientationonyourthumb.Infact,thechanceofitdoingthatisapproximately51%.Theiranalysisassumesthatthecoindoesn’tbouncewhenithitstheground,whichisreasonableforgrass,orwhenitiscaughtinthehand,butnotifitlandsonawoodentabletop.

The51%biasbecomesstatisticallysignificantonlyafterabout250,000flips.Itarisesbecausetheaxisaboutwhichthecoinisspinningmightnotbehorizontal.Asanextremecase,supposethattheaxisisatrightanglestothecoin,sothecoinalwaysremainshorizontalasitspins,likeapotter’swheel.Inthiscase,itwillalwayslandthesamewayupasitwastobeginwith,a100%chanceofnotflippingover.Theotherextremecaseiswhenthespinaxisishorizontal,andthecoinflipsendoverend.Althoughinprinciplethefinalstateisthendeterminedbytheupwardvelocityandrateofspinwhenthecoinleavesthehand,evensmallerrorsinspecifyingthesenumbersimplythatthecoinlandsthesamewayupasitstartedonly50%ofthetime.Withthistypeoftoss,thepredeterminedstateofamechanicalcoinisrandomisedbysmallerrors.

Usually,thespinaxisisinneitheroftheseextremelocations,butsomewhereinbetween,andclosetothehorizontal.Sothereisaslightbiasinfavouroflandingthesamewayup.Detailedcalculationsleadtothe51%figure.Experimentswithacoin-tossingmachineconfirmthisfigurereasonablywell.

Inpracticearealcointossisrandom,witha50%chanceofheadsortails,fornoneofthesereasons.Itisrandombecausetheinitialorientationofthecoin,whenit’ssittingonthethumb,israndom.Inthelongrun,thecoinstartsheadsuphalfthetime,andtailsuphalfthetime.Thisremovesthe51%biasbecausetheinitialstateisnotknownwhenthecoinistossed.

Howheads(white)andtails(shaded)varywiththeinitialrateofrotation(verticalaxis)andthetimespentintheair(horizontalaxis)whenthespinaxisishorizontal.Thehead/tailstripesbecomeverycloselyspaced

whenthespinrateishigh.

Seepage280forfurtherinformation.

PlayingPokerbyPostSupposethatAliceandBob—thetraditionalparticipantsinanycryptographicexchange—wanttoplaypoker,specifically,five-cardstud.ButAliceisinAliceSprings,Australia,whileBobisinBobbington,atowninStaffordshire,England.Cantheyperhapsmaileachotherthecards?Themainproblemisdealingthecards,a‘hand’offivetoeachplayer.Howcanbothplayersbesurethateachhasahandfromthesamepack,withouttheotherknowingtheirhand?

IfBobjustmailsAlicefivecards,shecan’tbesurethathehasn’tseenhercards;moreover,whenBobplayscardsfromwhatallegedlyishishand,shecan’tbesurewhetherhereallyhasonlyfivecardstoworkwith,orwhethertheremainderofthepackisavailabletohim,andheisonlypretendingtouseafixedhandoffivecards,dealtbeforethegamestarted.

Ifthishandarrivedinthepost,youcouldprobablybesurethedealerwasn’tcheating.Butformosthands,howcanyoutell?

Surprisingly,itispossibletoplayacardgamelikepokerbymail,oroverthephone,orovertheInternet,withoutanydangerthateitherplayerischeating.AliceandBobcanusenumbertheorytocreatecodes,andresorttoacomplicatedseriesofexchanges.Theirmethodisknownasazeroknowledgeprotocol,awaytoconvincesomeonethatyoupossessaspecificitemofknowledgewithouttellingthemwhatitis.Forexample,youcouldconvinceanonlinebankingsystemthatyouknowthesecuritycodeonthebackofyourcreditcard,withoutconveyinganyusefulinformationaboutthecodeitself.

Hotelsoftenlockguests’valuablesinasafetydepositboxinthereceptionarea.Toensuresecurity,eachboxhastwokeys:onekeptbythemanagerandtheotherbytheguest.Bothkeysareneededtoopenthebox.AliceandBobcan

useasimilaridea:

1Alicelocksacardineachof52boxes,usingpadlockswhosecodesonlysheknows.ShemailsthelottoBob.

2Bob(whocannotunlocktheboxestoseewhatcardsareinside)picksfiveboxesandmailsthembacktoAlice.Sheunlocksthemandreceivesherfivecards.

3Bobchoosesanotherfiveboxes,andputsanextrapadlockoneach.Heknowsthecodesforunlockingthese,butAlicedoesnot.HemailstheboxestoAlice.

4Aliceremovesherpadlocksfromtheseboxes,andmailsthembacktoBob.Hecannowopenthemtoreceivehisfivecards.

Afterthesepreliminaries,thegamecanstart.Cardsarerevealedbymailingthemtotheotherplayer.Toprovenoonecheated,theycanunlockalltheboxesaftertheendofthegame.

AliceandBobconvertthisideatomathematicsbyextractingtheessentialfeatures.Theyrepresentthecardsbyanagreedsetof52numbers.Alice’spadlockscorrespondtoacodeA,knownonlytoAlice.Thisisafunction,amathematicalrule,thatchangesacardnumbercintoanothernumberAc.(I’mtakinglibertieswiththenotationbynotwritingA(c),inordertoavoidtalkingabout‘composing’functions.)AlicealsoknowstheinversecodeA–1,whichdecodesAcbackintoc.Thatis,

A–1Ac=c

Bobdoesn’tknowAorA–1.Similarly,Bob’spadlockscorrespondtocodesBandB–1,knownonlyto

Bob,suchthat

B=1Bc=c

Withthesepreliminaries,themethodcorrespondstothepadlockprocedurelikethis:

1Alicesendsall52numbersAc1,…,Ac52toBob.Hehasnoideawhichcardsthesecorrespondto;ineffect,Alicehasshuffledthepack.

2Bob‘deals’fivecardstoAliceandfivetohimself.HesendsAlicehercards.Tosimplifythenotation,considerjustoneofthese,andcallitAc.AlicecanfindcbyapplyingA–1,sosheknowswhatcardsareinherhand.

3Bobneedstofindoutwhathisfivecardsare,butonlyAliceknowshowtoworkthatout.Buthecan’tsendhiscardstoAlicebecausethenshewillknowwhattheyare.SoforeachcardAdinhishand,heapplieshisowncodeBtogetBAd,andsendsthattoAlice.

4AlicecanagainapplyA–1to‘removeherpadlock’,butthistimethere’sasnag:theresultis

A–1BAd

InordinaryalgebrawecouldswapA–1andBroundtoget

BA–1Ad

whichequals

Bd

ThenAlicecouldsendthatbacktoBob,whowouldthenapplyB–1tofindd.However,functionscan’tbeswappedlikethis.Forexample,ifAc=c+1(so

A–1c=c–1)andBc=c2,then

A–1Bc=Bc–1=c2–1

whereas

BA–1c=(A–1c)2=(c–1)2=c2–2c+1

whichisdifferent.Thewayroundthisobstacleistoavoidthatsortoffunction,andsetupthe

codessothatA–1B=BA–1.Inthiscase,AandBaresaidtocommute,becausealittlealgebraturnsthisintotheequivalentconditionAB=BA.Noticethatinthephysicalmethod,Alice’sandBob’spadlocksdoindeedcommute.Theycanbeappliedineitherorder,andtheresultisthesame:aboxwithtwopadlocks.

AliceandBobcanthereforeplaypokerbypostiftheycansetuptwocommutingcodesAandB,sothatthedecodingalgorithmA–1isknownonlyto

Alice,andB–1isknownonlytoBob.BobandAliceagreeonalargeprimenumberp,whichcanbepublic

knowledge.Theyagree52numbersc1,…,c52(modp)torepresentthecards.Alicepickssomenumberabetween1andp–2,anddefineshercode

functionAby

Ac=ca(modp)

Usingbasicnumbertheory,theinverse(decoding)functionisoftheform

A–1c=ca’(modp)

foranumbera’thatshecancompute.Alicekeepsbothaanda’secret.BobsimilarlychoosesanumberbanddefineshiscodefunctionBby

Bc=cb(modp)

withinverse

B–1c=cb’(modp)

foranumberb’thathecancompute.Hekeepsbothbandb’secret.ThecodefunctionsAandBcommute,because

ABc=A(cb)=(cb)a=cba=cab=(ca)b=B(ca)=BAc

whereallequationshold(modp).SoAliceandBobcanuseAandBasdescribed.

EliminatingtheImpossibleFromtheMemoirsofDrWatsup

“Watsup!”“Uh—I’mnotsure,Soames.Whatisup?”“I’mcallingyourname,man,notaskingaquestion!HowmanytimeshaveI

toldyounottobringcopiesofTheStrandmagazineintothishouse?”“But—how—”“Youknowmymethods.Youweretappingyourfingersimpatiently,asyou

dowhenwaitingformetogoout.Andyoureyeskeptflickingtowardsthenewspaperrolledupinyourcoatpocket.WhichistoothickfortheDailyReporter,despitewhatitsaysonthefrontpage,soitmustcontainamagazine.Sinceyouhabituallyconcealonlyonesuchfromme,itsidentitywasneverindoubt.”

“I’msorry,Soames.Iwasjusthopingtogainsomecomparativeinsightsintoinvestigativemethodsfromthewritingsofthecompanionofthe,er,charlatanlivingacrossthestreetfromus.”

“Pah!Themanisafraud!Amountebankwhocallshimselfadetective!”TherearetimeswhenSoamescanbeoverbearing.Indeed,fewwhenheis

not,nowthatIcometothinkofit.“Ihaveoccasionallysiftedsomeusefulhintsfrommyexploitedcounterpart’soutpouringsofvapidity,Soames,”Iobjected.

“Suchas?”heasked,inanaggressivetone.“Iamimpressedbyhisargument:‘Whenyouhaveeliminatedthe

impossible,thenwhateverremains,howeverimprobable,mustbe—’““—wrong,”Soamesbrusquelycompletedmysentence.“Ifwhatremainsis

trulyimprobable,thenyouhavealmostsurelymadesomeunstatedassumptionwhendeclaringotherexplanationstobeimpossible.”

ConsistencyisnotoneofSoames’svirtues.“Well,perhaps,but—”“Nobuts,Watsup!”“Butonotheroccasionsyouhaveagreed—”“Pah!Realityisnotimprobable,Watsup.Itmaylookthatway,butits

probabilityis100%,becauseithashappened.”“Yes,technically,but—”“Acaseinpoint.Thismorning,Watsup,whileyouwereoutbuyingthat

scurrilousrag,Ireceivedanunexpectedvisitor.TheDukeofBumbleforth.”

“ThetoastofLondon,”Isaid.“Anoblemanofunimpeachableprobity,arolemodelforusall.”

“Indeed.Yetheinformedme…Well,therehad,hesaid,beenadinnerpartyatBumbleforthHall,atwhichtheEarlofMaunderingattemptedtoentertaintheguestsbyarrangingtenwineglassesinarowandfillingthefirstfiveofthem—likethis.”

Hedemonstratedwithourownglasses,fillingthemwitharatheracidMadeirathatwehaddecidedtothrowout.“Hethenchallengedthegueststorearrangetheglassessothattheyalternatedfullandempty.”

“Butthatiseasy,”Ibegan.“Ifyoumovefourglasses,yes.Interchangethesecondwiththeseventhand

thefourthwiththeninth.Likeso.”(Seethefigurebelow.)“However,theEarl’schallengewastoobtainthesameresultbymovingonlytwooftheglasses.”

Howtosolvethepuzzleinfourmoves

Ipressedmyfingerstogetherinanattitudeofdeepthought,andafteramomentIdrewaroughsketchoftheoriginalandfinalarrangements.“ButSoames:thefourglassesyouhavenamedmustallendupindifferentlocations!Soallfourmustmove!”

Henodded.“So,Watsup,youhavenoweliminatedtheimpossible.”“ByJove,yesIhave,Soames!Incontrovertibly.”Hebeganstuffingtobaccointohispipe.“SowhatdoyouconcludeifItell

youthataccordingtotheDukeofBumbleforth,afteralloftheguestshadexpressedsimilaropinions,theEarlofMaunderingthendemonstratedasolution.”

“I—uh—”“YouareforcedtoconcludethatthehonourableDuke,ascionoftheBritish

Empireandamanofhighnobility…isactuallyabaseliar.Fornosolutionexists,asyouhaveproved.”

Myfacefell.“Itdoesseem—no,wait,perhapsyouarenottellingmethe—”“Mydeardoctor,IfreelyconfessthatIdissemblefromtimetotime,always

withyourownbestinterestsatheart,butnotonthisoccasion.Youhavemyword.”

“Then…IamshockedattheDuke’sbehaviour.”“Comenow,Watsup,HavefaithinBritishcharacter.”“TheEarlcheated?”“No,no,no.Nothingofthekind.Youcandobetterthanthis.Theremaybe

anotherperfectlyprosaicexplanationthatyouhaveoverlooked.Infact,Ipredictthatyouwillshortlybetellingmehowchildishlysimpletheansweris.”

SoamesthentoldmewhatMaunderinghaddone.“Why,howchildishlysim—”Ibegan.Istoppedabruptly,andamforcedon

groundsofcandidnesstoadmitthatIflushedadeepcrimson.

WhatwasMaundering’ssolution?Seepage281fortheanswer.

MusselPowerIt’sanidyllicseasidescene:aquietbaywithwavesbreakingovertherocks,whicharefestoonedwithclumpsofshellfishandseaweed.Butthosesedatestaticmusselbedsareactuallyahiveofactivity.Toseeit,youjusthavetospeeduptheflowoftime.Intime-lapsephotography,themusselsareconstantlyonthemove.Theytetherthemselvestotherocksusingspecialthreads,secretedbytheirfoot.Bydetachingsomethreadsandaddingothersinnewlocations,themusselscancontroltheirpositionsontherocks.Ontheonehand(foot?),theyliketostayclosetoothermusselsbecausethatwaytheyarenotsolikelytoberippedofftherockbythewaves.Ontheother,theygetmorefoodiftherearen’tothermusselsnearbytocompetewiththem.Facedwiththisdilemma,musselsdowhatanysensibleorganismwould:theycompromise.Theyarrangethemselvessothattheyhavealotofnearneighboursbutfewdistantones.Thatis,theycongregateinpatches.Youcanseethepatcheswiththenakedeye,butnothowtheyform.

Clumpofbluemussels

In2011MoniquedeJagerandco-workersappliedthemathematicsofrandomwalkstodeducehowthemussels’clumpingstrategymighthaveevolved.Arandomwalkisoftencomparedtoadrunkardmovingalongapath:sometimesforward,sometimesback,withnoclearpattern.Goingupadimension,arandomwalkintheplaneisaseriesofsteps,whoselengthsanddirectionsarechosenrandomly.Differentrulesforthechoices—differentprobabilitydistributionsforthelengthsanddirections—leadtorandomwalks

withdifferentproperties.InBrownianmotion,thelengthisdistributedinabellcurve,closetoonespecificaveragestepsize.InaLévywalk,theprobabilityofmakingastepisproportionaltosomefixedpowerofitssize,somanyshortstepsareoccasionallyinterruptedbyamuchlongerone.

StatisticalanalysisofobservedstepsizesshowveryclearlythataLévywalkfitswhatmusselsonintertidalmudflatsactuallydo,whereasBrownianmotiondoesn’t.Thisagreeswithecologicalmodels,whichdemonstratemathematicallythataLévywalkdispersesthemusselsfaster,opensupmorenewsites,andavoidscompetitionwithotherspeciesofshellfish.Thisinturnsuggestswhythisparticularstrategymayhaveevolved.Naturalselectionprovidesafeedbackloopbetweenmovementstrategiesandthegeneticinstructionsthatcausethemtobeused.Individualmusselsaremorelikelytosurviveiftheyemploystrategiesthatincreasetheirchancesofobtainingfood,anddecreasetheirchancesofbeingsweptawaybyawave.

DeJager’steamusedfieldobservationsofwhatmusselsdoandsimulationsofmathematicalmodelsoftheevolutionaryprocess.ThesimulationsshowedthatLévywalksarelikelytoevolveasaresultofthispopulation-levelfeedback,andtheconditionsforthatstrategytobeevolutionarilystable—thatis,notsusceptibletoinvasionbyamutantwithadifferentstrategy—predictthatthepower-lawexponentshouldbe2.Thefielddataindicateavalueof2.06.

Thenovelfeatureofmusselbeds,inthiscontext,isthattheeffectivenessofanindividual’smovementstrategydependsonwhatalltheothermusselsaredoing.Eachmussel’sstrategyisdeterminedbyitsowngenetics,butthesurvivalvalueofthatstrategydependsonthecollectivebehaviouroftheentirelocalpopulation.Sohereweseehowtheenvironment—intheformoftheothermussels—interactswithindividualgenetic‘choices’toproducepatternformationonthepopulationlevel.

Seepage281forfurtherinformation.

ProofThattheWorldisRoundMostofusareawarethatourplanetisround—notanexactsphere,though:abitflattenedatthepoles.Ithasenoughbumpstoturnitintoapotatoifyouexaggeratethediscrepancyfromaspheroidbyafactorofabout10,000.Afew—veryfew—hardysoulspersistinthebeliefthattheworldisflat,eventhoughtheancientGreeks,2,500yearsago,amassedevidenceofitsrotunditythatconvincedevenmedievalclerics,andmoreevidencehasbeenpilingupeversince.BeliefinaflatEarthalmostdiedout,butitwasrevivedaround1883withthefoundingoftheZeteticSociety.ThisbecametheFlatEarthSocietyin1956,anditisstillactivetoday.YoucanfinditontheInternetandfollowitonFacebookandTwitter.

There’saneasyandvirtuallyfoolproofwaytocheckforyourselfthatourplanetcan’tbeflatiftheusualgeometryofEuclidapplies.ItrequiresInternetaccessoratoleranttravelagent,butnootherspecialapparatus,andit’snotlookinguptheshapeonWikipedia.Themethoddoesn’tofitselfshowthattheEarthisround,butasystematicandcarefulextensionwouldbeabletodojustthat.I’lldiscusspotentialwaystodenythisevidenceinamoment.Idon’tclaimthere’snowayout—ifyou’reaflat-Eartherthere’salwaysawayout.Butinthisinstance,thestandardploysareevenlessconvincingthanusual.Inanycase,theargumentmakesarefreshingchangefromtheusualscientificevidenceforaroundworld.

I’mnotthinkingofsatellitephotosofaroundplanet—thoseare,ofcourse,fakes.WeallknowthatNASAneverwenttotheMoon,itwasalldoneinHollywood,whichprovesthey’refakes,sothere.Nothingthatreliesonscientificmeasurement,either:thosescientisttypesarewell-knownhoaxers,theyevenpretendtheybelieveinevolutionandglobalwarming,bothofwhichareleftieplotstostopclean-living,righteous-thinkingpeoplemakingtheobscenesumsofmoneythataretheirGod-givenright.

No,whatIhaveinmindiscommercialevidence:airlineflighttimes.YoucanlooktheseupontheInternet:makesureyouuseactualflightsthatexist,notflight-timecalculatorsthatassumearoundEarth.

Forcommercialreasonsalllargepassengerjetsflyataboutthesamespeed.Iftheydidn’t,othercompanieswouldgetallthebusinessfromtheslowones.Theyflytheshortestroute,subjecttolocalregulations,forsimilarreasons.Sowecanusetraveltimesasreasonablyaccurateestimatesofdistances.(Toreduce

theeffectsofwind,takeasuitableaverageofflighttimesinbothdirections—inpracticetheusualarithmeticmeanisgoodenough,butSeepage281.)Thesurveyor’stechniqueoftriangulation,whichconstructsanetworkoftriangles,canthenbeusedtomapoutthelocationsoftheairportsconcerned.ForthepurposeofshowingthataflatEarthdoesn’twork,wecanassumeit’sflatandseewhatthatimplies.Surveyorsusuallyworkwithoneinitialdistance,thebaseline,andcalculateeverythingelsefromtheanglesofthetriangle,butwehavetheluxuryofusingactualdistances(inunitsofaircraft-hours).

Flat-Earthairlinemap

Thefigureshowsatriangulationbasedonsixmajorairports.Giveortakealittlejiggling,thisistheonlyplanararrangementthatisareasonablefittothetraveltimes.StartwithLondonandaddCapeTowndistance12away.AfterthatplaceRiodeJaneiroandSydney.Theirlocationsareunique,exceptthattheentiremapcouldbereflectedleft-rightwithoutchanginganydistances.Thatambiguitydoesn’tmatter,butyoudohavetoconfirmthatRiodeJaneiroandSydneyareonoppositesidesofthelinefromLondontoCapeTown.Iftheywereonthesameside,theflighttimebetweenthemwouldbeabout11hours,butactuallyit’s18.YoucanaddLosAngelesnext,andfinallylocateTahiti,againusingextratimingstoremoveambiguities.

NowwecanusethehypothesisofaflatEarthtomakeaprediction.ThedistancefromTahititoSydney,measuredfromthemap,isabout35hours.(Asithappens,therouteviaRioandCapeTownisveryclosetoastraightline,andthesumofthedistancesis35.)Sothat’stheminimumtimeitshouldtaketotravelbyplane,notcountingstops.

Theactualfigureis8hours.Evenallowingforminorerrors,thedifferenceiswaytoolarge,andthehypothesisofaflatplanetmustberejected.Withanetworkwithmanymoreairports,andmoreprecisefigures,youcouldmapoutthebasicshapeofmuchoftheplanetveryaccurately—stillinunitsofaircraft-hours.Tosetthescaleyou’dneedtoknowhowfasttheplanesfly,ormakeatleastonedistancemeasurementsomeotherway.

UnitedNationslogo:azimuthalequiangularprojectionfromroundEarthtoflatdisc

Now,everywell-informedflat-Eartherisawareofformsofwordsandnon-standardphysicsthatcan‘explain’theseresults.Perhapssomekindofdistortionfieldaltersthegeometry,sothattheliteralimageofaplanewithitsusualmeasureofdistanceiswrong.Thisreallydoeswork:anazimuthalequiangularprojectionoftheEarthfromthenorthpoledoesjustthat,andyoucantransfereverythingfromaroundEarthtoaflatone,lawsofphysicsincluded,usingprojectionontoaflatdisc.Providedyoumissouttheregionaroundthesouthpole.TheUnitedNationslogodoesjustthat,andhasbeenusedbytheFlatEarthSocietyto‘prove’thatitsviewsarecorrect.However,thiskindofchangeistrivialandmeaningless.It’salogicallyequivalentmodeltoaroundEarthwithconventionalgeometry.Mathematicallyitisjustalessthancandidwaytoadmit‘it’snotflat’,withintheorthodoxmeaningofthatphrase.Soalteredmetricsandsimilarexcusesreallydon’thackit.

Effectsofwind?Maybethere’sareallyhighwindblowingfromTahititoSydney?Itwouldhavetoblowat750mph,butworsethanthat:thestraightlinepathbetweenTahitiandSydneyisveryclosetotheroutesTahiti-Rio-CapeTown-Sydneythatwe’vealreadytakenintoaccount.IfyoucouldgetfromTahititoSydneyreallyquicklybyexploitingthewind,atleastoneofthoseroutesistakingwaytoolong.

Thenextlineofdefencewouldbethestandarddenialistploy:it’sallahugeconspiracy.Yes,butbywhom?ThetimeslistedonInternetbookingsitescan’tbefarwrong,becausemillionsofpeopletravelbyaireverydayandmostofthemwouldnoticeifthescheduledtimeswerefrequentlywildlywrong.Buttheairlinesmightallbeconspiringtoflymoreslowlythannecessaryonsomeroutes,sothatmostofmymapshouldshrink,makingitpossibletogetfromTahititoSydneyinamere14hours.Youhavetodivideby4ormoretomakethispossible,soaconventionalpassengerjetcouldactuallygetfromLondontoSydneyin5hoursiftheairlinewasn’tdeliberatelydawdlingtoconvinceusthe

planetisround.Unlikeallegationsofscientistsconspiring,whichonlymakesensetopeople

whodon’tknowanyscientists,*thisonehasaflawthatisprettymuchfatal.Itrequiresmostairlinestolosehugeamountsofmoneyeverydayinwastedfuel,andtorefrainfromwipingoutthecompetitionbyflyingroutesinlessthanhalfthetimetheycurrentlytake.AconspiracytomaketheEarthappearround,usingtheairline-schedulemetric,wouldrequirehundredsofprivate-sectorcompaniestovoluntarilythrowawayvastsumsofmoney.Areyoumad?

Youcan,ofcourse,alwaysfallbackonthatoldstandby:whenallelsefails,ignoretheevidence.

Footnote

*I’mnotreferringtothepropositionthatscientistsaremostlyhonest.I’mreferringtothedelighttheyalltakeinprovingeachotherwrong,whichamongotherthingsishowtheygetpromoted.Massiveconspiracieswouldstillmakenosenseevenifallscientistswerecrooks.

123456789TimesXContinuedThere’snoneedtostopat9(Seepage115).Trymultiplying123456789by10,11,12,andsoon.Whatdoyounoticenow?

Seepage282fortheanswer.

ThePriceofFame

WładysławOrlicz

WładysławRomanOrlicz,aPolishtopologist,introducedwhatarenowknownasOrliczspaces.Thesearehighlytechnicalconceptsinfunctionalanalysis.Onedayhisfameprovedcounterproductive.Likemostofhiscompatriots,helivedinaverysmallapartment,andonedayheappliedtothecityofficialsaskingforabiggerone.Thereplywas:“Weagreethatyourapartmentisverysmall,butwemustdenyyourclaimsinceyouhaveyourownspaces.”

TheRiddleoftheGoldenRhombusFromtheMemoirsofDrWatsup

Thespectacularsuccessofourjointendeavoursencouragedmeoncemoretotakeupmedicalpractice,andIarrangedforasmallsurgerytobeconstructedinmyhouse.ButItookcaretoallowsufficientflexibilityforoccasionswhenSoamesmightrequiremyservices,withorwithoutadequatenotice.Sowhenthetelegramarrived,IhandedmypatientovertomylocumDrJekyll,andsummonedacabtotakemeto222BBakerStreet.

WhenIarrivedatSoames’slodgingsIfoundhimsurroundedbypiecesofpaper.Inhishandswasapairofscissors.

“Aprettypuzzle,”heremarked.“Merelyarectangularstripofpaper,tiedintoasimpleoverhandknot.Itishardtoimaginethataman’sfatemaydependuponit.”

Knottedstripofpaper

“GreatHeavens,Soames!Howcouldthatbe?”“Anastycaseofextortion,Watsup.Theevidencehingesontheshapethatis

formedwhentheknotispulledastightlyaspossible,andflattenedout.Isuspectthatitwillturnouttobethesymbolofasecretsociety,andifIcansoprove,mycasewillbecomplete.”Heheldtheknotbeforemyeyes.“So,Watsup,whatshapewillwesee,eh?”

Iquicklysketchedanoverhandknotinmynotebook.

Overhandknottiedinaclosedloopofstring

“Itiswellknownthatwhenanoverhandknotistiedinaclosedloop,ithasthreefoldsymmetry,”saidI,feelingunusuallyastute.“Iwouldthereforeexpecteitheratriangleorahexagontoform.”

“Letustrytheexperiment,then,”saidSoames.“Andthenweshalltacklethemoredifficulttaskofprovingthatoureyesdonotdeceiveus.”

Whatshapeistheflattenedknot?Tryit.Seepage282fortheanswerandaproof.

APowerfulArithmeticSequenceAnarithmeticsequence(asequenceofnumberswithconstantdifferences)ispowerfulifthesecondtermisasquare,thethirdtermacube,andsoon.Thatis,thekthtermisakthpower.(Thisimposesnoconditiononthefirsttermsinceallnumbersarefirstpowersofthemselves.)Forexample5,16,27haslength3andcommondifference11,and

5=5116=4227=33

Atrivialwaytogetapowerfulsequenceoflengthnistorepeatntimesthenumber2n!.Thisisafirstpower,asquare,acube,andsoon,uptoannthpower.Thecommondifferenceiszero.

In2000JohnRobertsonprovedthatexcludingsequenceslikethis,wherethesamenumberrepeats—commondifferencezero—thelongestpossiblepowerfulsequencehasfiveterms(length5).SeeJohnP.Robertson,Themaximumlengthofapowerfularithmeticprogression,AmericanMathematicalMonthly107(2000)951.Toobtainthissequence,startwiththenumbers1,9,17,25,33,whichformanarithmeticsequencewithcommondifference8,andmultiplyeachofthemby32453011241720.Theresultingnumbersalsoformanarithmeticsequence,withcommondifference8timesthisnumber.Theyare:

(1)10529630094750052867957659797284314695762718513641400204044879414141178131103515625

(2)94766670852750475811618938175558832261864466622772601836403914727270603179931640625

(3)179003711610750898755280216553833349827966214731903803468762950040400028228759765625

(4)263240752368751321698941494932107867394067962841035005101121985353529453277587890625

(5)347477793126751744642602773310382384960169710950166206733481020666658878326416015625

Thecommondifferenceis

84237040758000422943661278378274517566101748109131201632359035313129425048828125000

Ifthefivetermsarea1,a2,a3,a4,a5,then

a1isthefirstpowerofitself(obviously)a2=3078419575898491388288844129170837402343752isasquarea3=56357797471169485761035156253isacubea4=7162889984611066406254isafourthpowera5=510722993555156255isafifthpower

Wow!(It’seasiertocheckthatthetermsarethestatedpowersifyouworkwiththe

primefactors.)

WhyDoGuinnessBubblesGoDownwards?Anyonewhodrinksdarkstout,suchasGuinness,willhaveseensomethingthatappearstoflyinthefaceofconventionalphysics.Thebubblesinthebeermovedownwards.Atleast,theyseemto.Butbubblesarelighterthanthesurroundingfluid,sotheyexperienceabuoyancyforcethatpushesthemup.

It’sagenuinemystery,oratleastitwasuntil2012whenateamofmathematicianssolvedit.Appropriately,theywereIrish(orbasedinIreland):WilliamLee,EugeneBenilov,andCathalCumminsoftheUniversityofLimerick.

Thesameeffectoccursinotherliquids,butit’seasiertoseeinstoutbecauselight-colouredbubblesshowupmoreclearlyagainstadarkbeer.Itisenhancedbecausestoutbubblescontainnitrogenaswellasthecarbondioxidethatisfoundinallbeers,andnitrogenbubblesaresmallerandlastlonger.

Partoftheansweriseasy:we’reonlyseeingthebubblesthatareneartheglass.Thoseoutinthemiddlearehiddenfromviewbythedarkbeer.Somaybesomebubblesaregoingup,butothersdown.Whatthatfailstoexplainiswhyanybubblesgodown.Theyshouldn’t.

Untilafewyearsagowedidn’tevenknowwhetherthewholethingwasanopticalillusion.Onealternativeexplanationisthattheeffectiscausedbydensitywaves—regionswherebubblesbunchup.Thebubblesgoupwardsbutthedensitywavesgotheotherway.Thiskindofbehaviouriscommoninwaves.Forexample,thewaterinoceanwavesdoesnottravelalongwiththewave;itmostlygoesroundandroundinroughlythesameplace.Whatmovesisthelocationofthehighestpartsofthewater.Admittedly,wavesbreakingonabeachdogoupthebeach;however,someofthatistheeffectofshallowwater,andthewaterrunsbackdowntothesea.Ifthewaterweretravellingwiththewaves,itwouldhavetopileupeverhigheronthebeach,whichdoesn’tmakemuchsense.Althoughthewaterdoesn’tgobackwardstoanysignificantextent,thisfamiliarexampleshowsthedifferencebetweenwherethewatergoesandwherethewavesgo.Nowdoitwithbubbles.

It’safairlyplausibletheory,butin2004agroupofScottishscientistsledbyAndrewAlexander,workingwithcolleaguesinCalifornia,producedvideofootageprovingthatthebubblesreallydomovedownwards.TheteamreleaseditsresultsonStPatrick’sDay.Theyusedahigh-speedcameratoslowdownthemovementandtrackindividualbubbles.Theyfoundthatthebubblestouching

thewallsoftheglasstendedtostick,sotheycouldn’tmoveupwards.However,bubblesnearthemiddlewerefreetorise,whichcausedthebeertoflowupinthemiddleanddownatthesides,draggingthebubbleswithit.

FlowofGuinnessinaglass:downattheedges

TheIrishteamrefinedthisexplanation,showingthatit’snotcausedbybubblesstickingtothewalls.Whatmakesithappenistheshapeoftheglass.Stoutisusuallydrunkfromaglasswithcurvedsides,whichiswideratthetopthanatthebottom.Usingfluiddynamicscalculationsandexperiment,theteamfoundthatwhenthebubblesnearthewallrise,theygostraightup,asyou’dexpect.Butthewallslopesawayfromthevertical,soineffectthebubblesmoveawayfromthewall.Thebeernearthewallisthereforedenserthanthatinthemiddle,soittendstoslidedownthesideoftheglass,draggingthenearbyfluidwithit.Sothebeercirculates:upwardsinthemiddle,downwardsnearthesides.

Thebubblesarealwaysgoingupwardsrelativetothebeer,butattheedgesthebeergoesdownfasterthanthebubblesgoup,sothebubblesgodowntoo.Weseethebubbles,butwecan’teasilyseethemotionofthebeer.

Seepage286forfurtherinformation.

RandomHarmonicSeriesTheinfiniteseries

iscalledtheharmonicseriesbymathematicians.Thenameislooselyrelatedtomusic,wheretheovertonesofavibratingstringhavewavelengths1/2,1/3,1/4,andsoon,ofthestring’sfundamentalwavelength.However,theserieshasnomusicalsignificance.Itisknowntobedivergent,meaningthatthesumuptontermsbecomesaslargeaswewishifnislargeenough.Itdivergesveryslowly,butitdoesdiverge.Infact,thefirst2ntermsadduptomorethan1+n/2.Ontheotherhand,ifwechangethesignofeveryotherterm,wegetthealternatingharmonicseries

whichconverges.Itssumislog2,whichisabout0.693.ByronSchmulandwonderedwhathappensifsuccessivesignsarechosenat

random,bytossingafaircoinandassigningaplussignto‘heads’andaminussignto‘tails’.Heprovedthat,withprobabilityone,thisseriesconverges(theharmonicserieswouldcorrespondtotossingHHHHHH…forever,whichhasprobabilityzero).However,thesumdependsonthesequenceoftosses.

Thequestionnowarises:whatistheprobabilityofgettingaparticularsum?Thesumcanbeanyrealnumber,positiveornegative,sotheprobabilityofgettinganyspecificnumberiszero(thisisgenerallythecasefor‘continuousrandomvariables’).Thewaytodealwiththisistointroduceaprobabilitydistribution(ordensity)function.Thisdeterminestheprobabilityofgettingasuminanygivenrangeofvalues,saybetweentwonumbersaandb.Thisprobabilityistheareaunderthedistributionfunctionbetweenx=aandx=b.

Forharmonicseriesmodifiedusingrandomcointosses,theprobabilitydistributionlookslikethefigurebelow.It’sabitlikethefamiliarbellcurve,ornormaldistribution,butthetoplooksflat.Ithasleft–rightsymmetry,correspondingtoswapping‘heads’and‘tails’onthesymmetriccoin.

Probabilitydistributionforrandomharmonicseries

Thisproblemisanobjectlessonin‘experimentalmathematics’,inwhichcomputercalculationsareemployedtosuggestinterestingconjectures.Itlooksasthoughthecentralpeakisatheight0.25,thatis,1/4.Italsolooksasthoughthevaluesofthefunctionat–2and+2are0.125,thatis,1/8.In1995KentMorrisonconjecturedthatbothstatementsaretrue,butin1998hechangedhismind,havinginvestigatedtheconjecturesinmoredetail.Totendecimalplaces,thedensityhasvalue0.2499150393atx=0,slightlylessthan1/4.However,totendecimalplacesthevalueatx=2is0.1250000000,whichstilllookslike1/8.To45decimalplaces,however,thevalueturnsouttobe:

0.124999999999999999999999999999999999999999764

whichdiffersfrom1/8bylessthan10–42.Schmuland’spaper[Randomharmonicseries,AmericanMathematical

Monthly110(2003)407–416]explainswhythisprobabilityissocloseto,butnotexactly,1/8.Sohereaveryplausibleconjecturefromexperimentalevidenceturnsouttobefalse.Thisiswhymathematiciansinsistonproofs,justasHemlockSoamesinsistsonevidence.

TheDogsThatFightintheParkFromtheMemoirsofDrWatsup

TakingmyusualmorningconstitutionalinEquilateralPark,justoffMaryleboneRoadneartheDogandTrianglepublichouse,Iobservedacuriousincident,andonarrivingto222BBakerStreetIcouldnotrestrainmyselffromsharingitwithmycolleague.

“Soames,Ihavejustwitnessedacurious—”“Incident.Yousawthreedogsinthepark,”saidhe,notbattinganeyelid.“Buthow—ofcourse!Thereismudonmytrousers,andthespatterpatterns

indicate—”Soameschuckled.“No,Watsup,mydeductionhasanotherbasis.Ittellsme

notonlythatyousawthreedogsinthepark,butthattheywerefighting.”“Sotheywere!Butthatwasnotthecuriousincident.Itwouldhavebeen

curioushadthedogsnotfought.”“True.Imustrememberthatremark,Watsup.Mostapposite.”“Whatwascuriouswaswhatprecededthefight.Thedogsappeared

simultaneouslyatthethreecornersofthepark—”“Whichisanequilateraltrianglewhosesidesareall60yards,”Soames

interjected.“Uh,yes.Atthemomenttheyappeared,eachdogfacedthenextina

clockwisedirection,andimmediatelybegantoruntowardsit.”“Eachatthesamespeedof4yardspersecond.”“Ibowtoyourjudgement.Asaresult,allthreedogsfollowedcurvedpaths,

andcollidedsimultaneouslyatthecentreofthepark.Inaflash,theywerefighting,andIhadtoseparatethem.”

“Whencethetearsinyourcoatandtrousers,andtheteethmarksonyourleg,whichIseewereinflictedbyoneredsetter,oneretriever,andonecrossbetweenabulldogandanIrishwolfhound.Withalamefrontleftleg.”

“Ah.”“Andwearingaredleathercollar.Withabellonit.Whichhasrustedandno

longerrings.Wereyouobservantenoughtonoticehowlongittookthedogstocollide?”

“Ineglectedtolookatmypocket-watch,Soames.”“Oh,comenow,Watsup!Youlook,butyoudonotsee.However,inthis

casethetimecanbededucedfromthefactsalreadyestablished.”

Assumethedogsarepoints.Seepage286fortheanswer.

Thethreedogs

HowTallisThatTree?There’sanoldforester’strick(thetrickisold,nottheforester)forestimatingtheheightofatreewithoutclimbingitorusingsurveyingequipment.Itcanbeagreaticebreakeratgardenpartiesifthere’sasuitabletreeinthevicinity.IlearneditfromTobyBuckland,Diggingdeeper,AmateurGardening(20October2012)page59.Trousersaretherecommendedattire.

Standareasonabledistancefromthetree,withyourbacktowardsit.Bendoverandlookbackatitthroughyourlegs.Ifyoucan’tseethetop,moveawayuntilyoucan.Ifyoucanseeiteasily,movecloseruntilit’sjustvisible.Atthatpoint,yourdistancefromthebaseofthetreewillberoughlyequaltoitsheight.

Estimatingtheheightofatree

Thetechnique,ifImaycallitthat,isasimpleapplicationofEuclideangeometry.Itworksbecausetheangleatwhichmostofuscanlookupwardsthroughourlegsisroughly45˚.Sothelineofsighttothetopofthetreeisthehypotenuseofanisoscelesright-angledtriangle,andtheothertwosidesareequal.

Obviouslytheaccuracyofthemethoddependsonhowflexibleyourbodyis,butit’snottoofarwrongformanyofus.Bucklandremarks:“Haveago,it’scheaperthanyogaandoffersaviewontheworldthatmosthaven’tenjoyedsincechildhood!”

WhyDoMyFriendsHaveMoreFriendsThanIDo?OMG!EveryoneseemstohavemorefriendsthanIdo!

IthappensonFacebook,ithappensonTwitter.Ithappensonanysocialmediawebsite,andithappensinreallife.Ithappensifyoucountbusinesspartnersorsexualpartners.It’sahumblingexperiencewhenyoustartcheckingyourfriendstoseehowmanyfriendstheyhave.Notonlydomostofthemhavemorethanyoudo:onaverage,theyallhavemorethanyoudo.

Whyareyousounpopularcomparedtoeveryoneelse?It’sveryworrying.Butthere’snoneedtobeupset.Mostpeople’sfriendshavemorefriendsthantheydo.

Samplefriendshipnetwork

Thatprobablysoundsweird.Everyoneinagivensocialnetworkhasonaveragethesamenumberoffriends;namely,theaverage–thereisonlyone.Somehavemore,somehavefewer,buttheaverageis…theaverage.Itthenseemsintuitivelyplausiblethattheirfriendswill,onaverage,alsohavethatverysamenumberoffriends.Butisittrue?

Let’stryanexample.It’snotconcoctedtobeunusual;it’sthefirstoneIdrew.Mostnetworkswillbehavesimilarly.Thenetwork(above)shows12people,withlinesconnectingthefriends.(Weassumeallfriendshipsworkbothways.That’snotalwaysthecaseforsocialnetworks,buttheeffectstilloccursevenso.)Tabulateafewkeyfigures:

I’veusedboldfacetomarkthenumbersinthefinalcolumnthatarebiggerthanthenumbersinthesecond.ThesearethecaseswherepersonX’sfriendshave,onaverage,morefriendsthanpersonXhas.Eightoutoftwelvenumbersareboldface,andthere’sanotherwherethetwofiguresareequal.

Theaverageofthenumbersincolumn2is3.Thatis,theaveragenumberoffriends,acrosstheentiresocialnetwork,is3.Butthemajorityofentriesincolumn4arebiggerthanthat.Sowhat’swrongwithintuitionhere?

TheanswerispeoplelikeEthelandGwenwhohaveanunusuallylargenumbersoffriends,here5and6respectively.Becauseofthis,theygetcountedalotmoreoftenwhenwe’relookingathowmanyfriendspeople’sfriendshave.Andtheythencontributemoretothetotalincolumn3,hencetotheaverage.Ontheotherhand,peoplewithfewfriendsshowupmuchlessoften,andcontributeless.

Yourfriendsarenotatypicalsample.Peoplewithalotoffriendswillbeover-representedamongthem,becausethere’sagreaterchancethatyouareoneoftheirfriends.Peoplewithfewfriendswillbeunder-represented.Thiseffectskewstheaveragetowardsahighervalue.

Youcanseeithappeningincolumn3ofthetable.Thenumber5occursfivetimesincolumn3,oneforeachofEthel’sfriends;similarly6occurssixtimesincolumn3,oneforeachofGwen’sfriends.Ontheotherhand,Alice’scontributiontocolumn3(notinherrow,butwhensheoccursasafriendinotherrows)isjusttwo2’s:onefromBobandonefromCleo.SoEthelcontributes25,andGwenamassive36,whereaspooroldAlicecontributesjust4.

Tothemthathath,shallbegiven.Thisdoesnothappenforcolumn2:everyonecontributestheirfairshareto

theaverage,whichis3.Infacttheaverageofallthenumbersincolumn4is3.78,alotbiggerthan3.

Iprobablyoughttouseaweightedaverage:addallthenumbersincolumn3anddividebyhowmanyofthemthereare.Thisis3.33,againbiggerthan3.

Ihopeyoufeelhappiernow.

Seepage287foraproof.

Isn’tStatisticsWonderful?Statistically,42millionalligatoreggsarelaideveryyear.Ofthose,onlyhalfhatch.Ofthosethathatch,threequartersareeatenbypredatorsinthefirstmonth.Oftherest,only5%arealiveafterayear,foronereasonoranother.

Ifitwasn’tforstatistics,we’dallbeeatenbyalligators!

TheAdventureoftheSixGuestsFromtheMemoirsofDrWatsup

IthaslongdistressedmethatSoameshasaheartydislikeofdinnerparties.Hedespisessmalltalkandbecomesuneasyinthecompanyofwomen,especiallyattractivewomenlikemyfriendBeatrix.Butfromtimetotimeheisobligedtobitethebullet,graspthenettle,plaittheplatitude,andattendsocialeventsthatincludethefairersex.Atthese,hecanbefromonemomenttothenexttaciturn,obnoxious,charming,garrulous,orsomecombination.

Thisparticularoccasionwasamodesttête-à-têtewithAubreyandBeatrixSheepshear(brotherandsister)andCrispinandDorindaLambshank(husbandandwife).Iknewallfour,ofcourse;Beatrixisadelightfullady,unmarried,andwithoutacurrentsuitor,Ifirmlybelieve.Soamesknewonlyme,whichIfearedmightcausetheworsesideofhischaractertodominate,butIwashopingtowidenhissocialcircle.TheSheepshearsandLambshankshadnotmet,exceptforthemen,whobelongedtothesameclub.

AllthisbecamecleartoSoameswhentheguestsarrived,andsoonweweresittingtogether.Soames’spresencemadetheconversationspasmodic,soIventuredtopoursomeglassesofamodestbutacceptablesherry,givinghimadoubleshare.

“Howsingular!Iseebothatripleofmutualacquaintancesandatripleofmutualstrangers,”saidI,attemptingclumsilytobreaktheice.

“Atriplecannotbesingular,”Soamesmuttered,butatagesturefrommeheheldhispeace.Itoppeduphisglass.

Beatrixaskedmeforanexplanation,andIhastenedtocomply.“You,Aubrey,andIeachknowtheothertwo:atripleofmutualacquaintances.”

“Ithinkwearemorethanmereacquaintances,John,”shereplied.“Iamdelightedtohearit,dearlady,”saidI,“butIwasseekingawordthat

mightapplytoanypairofpeople.Incontrast,Soames,you,andDorindaaremutualstrangers,inthesensethatyouhavenotmetsociallyuntilnow.Ofcourse,Soames’sfamehasgonebeforehim.”

“Ithasindeed,”saidCrispin,givingmeasourlook.“Now,Ifindthisfactsomewhatremarkable—”“Youshouldnot,Watsup,”Soamesinterrupted.“Atleast,youshouldnot

consideritremarkablethatatleastonesuchtriple,acquaintancesorstrangers,

shouldoccur.”“Whynot?”askedAubrey.“Becauseatleastonesuchtriplemustoccurwheneversixpeopleare

gatheredtogether,”Soamesreplied.“Itmattersnotwhoknowswhom.”“Well,I’llbedashed,”saidAubrey.“Remarkable,what?”“Howcanyoubesosure,MrSoames?”Beatrixenquired,hereyesshining—

andnotjust,Isuspectedfromthesherry.“Because,mydearmadam,itcanbeproved.”“Oh.Dogoon,MrSoames.Ifindsuchmattersfascinating.”Soames

inclinedhishead,butIdetectedafaint,fleetingsmile.Hepretendstobeimmunetofemininecharms,butIknowthistobeasham.Hemerelylacksconfidence.Ihopedthiswouldcontinue,forBeatrixisprettyandmodestandwouldmakeagoodcatchforanycompatibleman.Myself,forexample.

“Theproofcanbeunderstoodmostsimplybywayofadiagram,”saidSoames.Herose,walkedacrosstothediningtable,andpickedupsomesideplatesandsomecutlery,brushingasidemyprotestationsalongwithseveralnapkins,themustard,andapottedaspidistra.

Soames’sdiagram“Theplatesrepresentthesixofus,”Soamessaid,writingourinitialsonthem

withastickofgreasepaint,whichhehadpresumablykeptasasouvenirfromthetimehehadcontemplatedacareeronthestage.“Aforkconnectingtwopeopledenotesthattheyareacquaintances;aknifedenotestheyarestrangers.”

“Lookingdaggersateachother,then,”saidBeatrix.Ihastenedtoapplaudherwitandrefillherglass.

“Forinstance,WatsupandIarejoinedbyaforkatthecentreofthetable,butIamjoinedtoeveryoneelsebyaknife.

“Now,asWatsuphassoastutelypointedout,WABisatriangleofforksand

SBDisoneofknives.Mycontention,however,isthathoweverwearrangeknivesandforks,therewillbeatleastonetriangleformedbyidenticaltypesofcutlery.”

“Coulditbeboth,MrSoames?”askedBeatrix.Hereyesfollowedhiseverymovement.

“Sometimes,madam,butnotalways.Togotoextremes,ifallimplementsareforksthenthereisnotriangleofknives,andiftheyareallknivesthenthereisnotriangleofforks.”

Beatrixnodded,aseriouslookonherface.“Itseems,then,”shemused,“thatasforksarereplacedbyknives,andtheopportunitytoformatriangleofforksdiminishes,thatofformingatriangleofknivesincreases.”

Soamesnodded.“Verywellput,madam.Theproofismerelyamatterofshowingthatthelattercomestopassbeforetheformerceases.Tobeprecise,letuschooseoneplate.Anyplate.Fiveimplementspointtowardsit.Atleastthreeofthemmustbeofthesamekind.Why?”

“Becauseifthereweretwoorfewerofeach,therewouldbeatmostfourimplements,”Beatrixsaidatonce.

“Verygood!”saidI,beforeSoamescouldofferasimilarcompliment.“Now,”saidhe,“letusconsiderasetofthreeidenticalimplements—letus

assumetheyareforks,theknifecaseissimilar—andlookattheplatestowhichtheypoint.Otherthantheoriginalchoice,ofcourse.Now,eitheroneofthoseplatesisjoinedtoanotherbyafork,or—”

“Allthreearejoinedbyknives!”shecried.“Inthefirstcase,wehavefoundatriangleofforks;inthesecond,oneofknives.Why,MrSoames,nowthatyouexplainitsoclearlyitseems—”

“Absurdlyobvious,”sighedSoames,takingalargesipofsherry.Thisremarksomewhatdeflatedher,andIwavedmyhandinapologyformy

friend’srudeness.Herinstantsmilewasmostgratifying.

ThisareaofmathematicsiscalledRamseyTheory.Seepage288forfurtherinformation.

HowtoWriteVeryBigNumbersHowmanygrainsofsandarethereintheuniverse?ThegreatestoftheancientGreekmathematicians,Archimedes,decidedtocombattheprevailingbeliefthattheanswerwasinfinite,byfindingawaytoexpressverylargenumbers.HisbookTheSandReckonerassumedthattheuniversewasthesizethatGreekphilosophersthoughtitwas,andthatitwasentirelyfilledwithsand.Hecalculatedthattherewouldbe(inourdecimalnotation)atmost1,000,…,000grainsofsand,with63successivezeros.

That’sbig,butnotinfinite.Aretherebiggernumbers?Mathematiciansknowthatthereisnolargest(whole)number.Theycanbe

asbigasyouwish.Thereasonissimple:iftherewerealargestnumber,youcouldmakeitlargerbyadding1.Mostchildrenwhohavemastereddecimalnotationquicklyrealisethatyoucanalwaysmakeanumberbigger(indeed,tentimesbigger)byputtinganextra0ontheend.

However,althoughthereisnolimitinprincipletohowlargeanumbercanbe,thereareoftenpracticallimitationsinthewaywechoosetowritenumbers.Forinstance,theRomanswrotenumbersusingthelettersI(1),V(5),X(10),L(50),C(100),D(500),andM(1,000),combiningthemingroupstogetintermediatenumbers.So1-4werewrittenI,II,III,IIII,exceptthatIIIIwasoftenreplacedbyIV(5minus1).Inthissystemthelargestnumberyoucanwriteis:

MMMMCMXCIX=4,999

andyoucanknockathousandoffthatifyoustopwiththreeM’s.However,theRomanssometimesneededbiggernumbers.Tosymbolisea

million,theyputabar(theirnamewasvinculum)overthetopofanM,toget .Ingeneral,abaroveranumbermultiplieditsvaluebyathousand,buttheyseldomusedthisnotation.Whentheydid,theyusedonlyonebar,soafewmillionwasasfarastheycouldget.Thelimitationsoftheirsymbolsystemshowthatthesizeofnumbersyoucanwritedowndependsonthenotationyouuse.

Nowadayswecangoalotfurther.Amillionis1,000,000—pathetic.Wecangetmuchbiggernumbersjustbyputtingmorezerosontheend,andmovingthecommasifnecessarytokeepthestandardgroupingintosetsofthreedigits.

(Mathematiciansgenerallyomitthecommasanyway,buttheyoftenreplacethembythinspaces:1000000.)IntheWesternworld,thestandarddictionarynamesforlargenumbersreflectthispractice:theystartwithmillion,billion,trillion,…andstopatcentillion.Sincehumanbeingsnevermanagetokeepthingssimple,especiallywhenitcomestomathematics,thesewordshave(oratleastusedtohave)differentmeaningsondifferentsidesoftheAtlantic.IntheUSAabillionis1,000,000,000,butintheUKitwas1,000,000,000,000—whichAmericanswouldcallatrillion.Butintoday’sinterconnectedworld,theAmericanusagehasprevailed,perhapsbecause‘milliard’—theUKtermforathousandmillion—is(a)obsolescentand(b)tooeasilyconfusedwith‘million’.Andabillionisaniceroundnumberforinternationalfinance,atleastuntiltheworld’sbanksthrewawaysomuchinthefinancialcrisisthatwehadtogetusedtothinkingintrillions.

Asimplerwaytowritethesenumbersistousepowersof10.So106denotes1followedbysix0’s,whichisamillion.The6iscalledanexponent.Abillionis109(or1012inold-fashionedUK-speak).Acentillionturnsouttobe10303(10600UK).Recognisedextensionstothestandarddictionarynamesgouptoamillinillion,103003.Thereareseveralsystemsandlifeistooshorttogointothemall,orindeeddistinguishproperlybetweenthem.

Twoothernamesforlargenumbers,alsofoundinmostdictionaries,aregoogolandgoogolplex.Agoogolis10100(1followedbyahundred0s);thenamewasinventedbyJamesNewman’snine-year-oldnephewMiltonSirotta.Sirottaalsosuggestedalargernumber,googolplex,whichhedefinedas‘1followedbywritingzerosuntilyougettired’.Acertainlackofprecisionledtoarefinement:‘1followedbyagoogolzeros’.

That’smoreinteresting,becauseitrunsintothesamesortofproblemthattheRomanshit,excepttheygottherealotsooner.Ifyouactuallytrytowriteagoogolplexdownindecimalnotation,1,000,000,000,…youwon’tgettotheendduringyourlifetime.Youwouldn’tgettoitduringthelifetimeoftheentireuniversetodate.Assumingthatconventionalcosmologicalcalculationsarecorrect,youprobablywouldn’tgettoitbeforetheuniverseended.Inanycase,you’drunoutofspacetowriteallthosezeros,evenifeachwerethesizeofaquark.

However,thereisacompactwaytowriteagoogolplex:iteratedexponents.Namely,

1010100

Andonceyoustartthinkingalongthoselines,youcangettosomeverybig

numbersindeed.In1976computerscientistDonaldKnuthinventedanotationforverylargenumbers,whichamongotherthingsturnupinsomeareasoftheoreticalcomputerscience.WhenIsay‘verylarge’Imeanverylarge—solargethatthere’snowayeventobegintowritethemdowninconventionalnotation.Thegoogolplex,whichis1followedby10100zeros,isdwarfedbymostofthenumbersyoucanexpressusingKnuth’sup-arrownotation.

Knuthstartsbywriting

a b=ab

So,forexample,10 2=100,10 3=1000,10 100isagoogol,and10 (10 100)isagoogolplex.Theusualconventionabouttheorderinwhichexponentsaretaken(startattherightandworkleftwards)letsuswritethismoresimplyas1010 100.Youdon’tneedalotofimaginationtocomeupwith10 10 10 10 10 1010,say.

Butthatisjustthestart.Let

a b=a a … a

whereaoccursbtimes.Theexponentialsareevaluatedstartingfromtheright,sothat(forexample)

a 4=a (a (a a))

Forexample,

2 4=2 (2 (2 2))=2 (2 4)=2 16=65,536

and

3 3=3 3 3=3 27=7,625,597,484,987

Thenumbersrapidlybecomeimpossibletowritedowndigitbydigit.Forinstance,4 4has155decimaldigits.Butthat’sthepoint up-arrownotationprovidesacompactwaytospecifygiganticnumbers.However,we’vebarelystarted.Nowlet

a b=a a … a

whereaoccursbtimesontheright-handside.Againthe sareevaluated

startingfromtheright.Yougetthepicture wecancontinuewith

a b=a a … aa b=a a … a

andsoon,whereasalwaysaoccursbtimesandweevaluatestartingfromtheright.

R.L.GoodsteindevelopedKnuth’snotationandsimplifiedit,leadingtoexpressionshecalledhyperoperators.JohnConwaydevelopedasimilar‘chainedarrow’notationwithhorizontalarrowsandbrackets.

Instringtheory,anareaoftheoreticalphysicsthataimstounifyrelativitywithquantummechanics,thenumber10 10 500turnsup itisthenumberofpotentiallydifferentstructuresforspace-time.AccordingtoDonPage,thelongestfinitetimeexplicitlycalculatedbyaphysicistisapaltry

10 10 10 10 10 1.1years

ThisisthePoincarérecurrencetimeforthequantumstateofablackholewiththemassoftheentireuniverse;thatis,thetimeitwouldtakebeforethissystemreturnedtoitsoriginalstateand,ineffect,historystartedtorepeatitself.

Graham’sNumberMathematiciansoccasionallyneedbiggernumbersthanphysicistsdo.Notjustforthefunofit:becausethesenumbersactuallyturnupinsensibleproblems.Graham’snumber,namedfortheAmericanRonGraham,arisesincombinatorics—themathematicsofcountingdifferentwaystoarrangeobjectsorfulfilconditions.

In1978GrahamandBruceRothschildwereworkingonaproblemabouthypercubes,multidimensionalanaloguesofthecube.Asquarehas4corners,acubehas8,afour-dimensionalhypercubehas16,andann-dimensionalhypercubehas2ncorners.Theycorrespondtoallpossiblesequencesofn0’sand1’sinasystemofncoordinates.

Takeann-dimensionalhypercube,anddrawlinesconnectingallpairsofcorners.Coloureachedgeeitherredorblue.Whatisthesmallestvalueofnforwhicheverysuchcolouringcontainsatleastonesetoffourcorners,lyinginaplane,suchthatalltheedgesjoiningthemhavethesamecolour?

Thetwomathematiciansprovedthatsuchanumbernexists,whichisfarfromobvious.Grahamhadearlierfoundasimplerproof,buthehadtousealargernumber:inKnuth’sup-arrownotationitisatmost:

Herethenumbersbelowthehorizontalbracesshowhowmanyarrowsoccurabovethebrace.Workbackwardsfromthebottomlayer:thereare3 3up-arrowsintheprevious(63rd)layer.Thenusethatnumberofarrowsinthe62ndlayertogetanewnumber.Thenusethatnumberofarrowsinthe61stlayer…!It’snotpossibletowritedownanyofthesenumbersinstandarddecimalnotation,sorry.Theyarefarworsethanagoogolplexinthatrespect.Butthat’stheircharm…

ThisisGraham’snumber,anditistrulygigantic.Andthensome.ThevaluefoundbyGrahamandRothschildissmaller,butstillridiculouslylarge,and

hardertoexplain,soIwon’t.Ironically,workersintheareaconjecturethatthenumbercanbemademuch

smaller.Infact,thatn=13willdo.Butthishasnotyetbeenproved.GrahamandRothschildprovedthatnmustbeatleast6;GeoffExooraisedthatto11in2003;thebestresultnowisthatnmustbeatleast13,provedin2008byJeromeBarkley.

Seepage289forfurtherinformation.

Can’tWrapMyHeadAroundItWhenscientistsmentionlargenumbers,suchastheageoftheuniverse(13.798billionyearsorabout4.35sextillionseconds)orthedistancetotheneareststar(0.237lightyearsorabout2.24trillionkilometres)wealltendtosaythingslike“Ican’twrapmyheadaroundthat”.Thesamegoesforthecostoftheglobalfinancialcrisis,oneofthehigherestimatesbeing£1.162trilliontotheUKeconomy.*Let’ssayaroundtrillion,£1012.

Millions,billions,trillions—inmanypeople’sminds,theseareallprettymuchthesame:toobigtowrapyourheadaround.

Thisinabilitytointernalisebignumbersaffectsourviewsonmanythings,especiallypolitics.Therewasmuchprotest,especiallyfromtheairlineindustry,whenEyjafjallajökullinIcelandspewedoutvolcanicashandgroundedmostoftheUK’saeroplanes.(Iwasn’ttoohappymyself:IwasduetoflytoEdinburghandhadtochangeplansrapidlyanddriveinstead.)Thecostwasestimatedat£100millionperday:£108.

Tobefair,thatwasthelosstoarelativelysmallnumberofcompanies.Butthescaleoftheoutcrywasprobablygreaterthanthatcausedbythefinancialcrisis.

Thebigsecretaboutcomparinglargenumbersisthatyoudon’thavetowrapyourheadroundthem.Indeed,itisprobablybestnotto.Themathematics—indeed,basicarithmetic—doesitforyou.Forexample,wecanaskhowlongtheflyingbanwouldhavetocontinueinordertocosttheeconomythesameamountthatthebankingcrisisdid.Thecalculationgoes:

Costofbankingcrisis:£1012Costperdayofvolcano:£1081012/108=104days=27years

Ifindthisperiodoftimeeminentlygraspable,andIhavenotroublerecognisingthatitisagreatdeallongerthanoneday.SoIcanworkoutthattheflyingbanwouldhaveneededtogoonfor27yearsbeforeitdidasmucheconomicdamageasthebankcrisis,withoutwrappingmyheadroundthelargerfiguresinvolvedinthecalculation.

Thisiswhatmathematicsisfor.Don’twrapyourheadroundthings:dothe

maths.

Footnote

*Thisfigurewasmorethanthefinalcostbecausethebankspaidmoneybackandsomeofitwastemporarysupport.ByMarch2011itwas£450billion,roughlyhalfasbig.

TheAffairoftheAbove-AverageDriverFromtheMemoirsofDrWatsup

Ithrewthenewspaperontothetableindisgust.“Isay,Soames—lookatthisridiculousstatistic!”

HemlockSoamesgrunted,andconcentratedonlightinghispipe.“Seventy-fivepercentofhansomcabdriversthinktheirabilityisabove

average!”Soameslookedup.“What’sridiculousaboutthat,Watsup?”“Well,I—Soames,it’sjustnotpossible!Theymusthaveanexaggerated

opinionofthemselves!”“Why?”“Becausetheaveragehastobeinthemiddle.”

Hansomcab from JohnThompson andAdolpheSmith,Street Life in London,1877

Thedetectivesighed.“Acommonmisconception,Watsup.”“Miscon—what’swrongwithit?”“Justabouteverything,Watsup.Suppose100peopleareassessedonascore

rangingfrom0to10.If99ofthemscore10andtheotherscores0,whatistheaverage?”

“Uh…990/100…whichis9.9,Soames.”

“Andhowmanyareaboveaverage?”“Uh…99ofthem.”“AsIsaid,amisconception.”Iwasnotsoreadilydiverted.“Buttheexcessissmall,Soames,andthedata

arenottypical.”“Iexaggeratedtheeffecttodemonstrateitsexistence,Watsup.Anydatathat

areskewed—asymmetric—arelikelytobehaveinasimilarmanner.Forexample,supposethatmostdriversarereasonablycompetent,asignificantminorityisappallinglybad,andaverytinynumberisexcellent.Whichdriversareaboveaverageinsuchcircumstances?”

“Well…thebadonesbringtheaveragedown,andtheexcellentonesdon’tcompensate…Myword!Thecompetentandexcellentdriversareallaboveaverage!”

“Indeed,”Soamesreplied.Herapidlysketchedagraphonasheetofscrappaper.“Withthesedata,whicharemorerealistic,theaverageis6.25,and60%ofthedriverslieabovethat.”

Soames’shypotheticaldrivingabilityscores,with60%ofdriversaboveaverage“SothearticleintheManchesterMirrographiswrong?”Ienquired.“Areyousurprised,Watsup?Veryfewofitsarticlesarecorrect,tobefrank.

Butthisonefallsintoacommontrap.Itconfusestheaveragewiththemedian—whichisdefinedtobethevalueforwhichhalfareaboveandhalfbelow.Thetwooftendiffer.”

“Soitisnotpossiblefor75%ofdriverstobeabovethemedian?”“Onlyifthenumberofdriversiszero.”“But75%ofdriverscouldbeaboveaverage?”“Yes.”“Andtheywouldn’thaveanundulyhighopinionoftheirownabilities?”Soamessighedagain.“That,mydearWatsup,isakettleofhorsesofa

differentcolouroffish.Thereisacommonformofcognitivebiascalledillusorysuperiority.Peopleimaginethemselvestobesuperiortoothers,evenwhentheyarenot.Almostallofussufferfromthisbias,withthenotableexceptionofmyself.AnarticleinQuantitativePhrenologyandCognitionlastmonthreportedthat69%ofSwedishcab-driversratedthemselvesabovethemedian.Andthatisdefinitelyillusory.”

Seepage289forgenuinemoderndata.

TheMousetrapCubeJeremiahFarrellinventedamagicwordcubeobeyingsimilarprinciplestothosegoverninghismagicwordsquares,Seepage20.HerethewordinvolvedisMOUSETRAPandthemagicnumberingofthelettersgoesM=0,O=0,U=2,S=6,E=9,T=18,R=3,A=1,P=0.Someofthewordsarepersonalnames,andothersareveryobscure.Forinstance,OSEisthenameofademon—andalsoofplacesinJapan,Nigeria,Poland,Norway,andSkye.Still,theamazingthingisthatitcanbedoneatall.

Successivelayersofthemousetrapmagicwordcube

SierpińskiNumbersNumbertheoristsseekinglargeprimesoftenconsidernumbersoftheformk2n+1,forspecificchoicesofk,asnvaries.Experimentsuggeststhatformostchoicesofk,thesenumbersincludealeastoneprime,oftenmore.Forinstance,ifk=1then1×2n+1isprimewhenn=2,4,8.Ifk=3then3×2n+1isprimewhenn+1,2,5,6,8,12.Ifk=5then5×2n+1isprimewhenn=1,3,7.(Ingeneral,wecandividekbyanyfactorsof2tomakeitoddandincludethesefactorsinthe2n.Sowemayassumekisoddwithoutlosinggenerality.Forexample,24×2n=3×23×2n=3×2n+3.)

Itistemptingtoconjecturethatforanyk≥2thereexistsatleastoneprimeoftheformk2n+1.However,in1960WacławSierpińskiprovedthatthereexistinfinitelymanyoddnumberskforwhichallnumbersoftheformk2n+1arecomposite.ThesearecalledSierpińskinumbers.

In1992JohnSelfridgeprovedthat78,557isaSierpińskinumber,byshowingthatallnumbersoftheform78,557×2n+1aredivisiblebyatleastoneofthenumbers3,5,7,13,19,37,73.Thesearesaidtoformacoveringset.ThefirsttenknownSierpińskinumbersare:

78,557271,129271,577322,523327,739482,719575,041603,713903,983934,909

Itiswidelybelievedthat78,557isthesmallestSierpińskinumber,butthathasnotyetbeenprovedordisproved.Since2002thewebsitewww.seventeenorbust.comhasbeenorganisingasearchforprimesoftheformk2n+1,whoseexistencewouldprovethatkisnotaSierpińskinumber.Whenthesearchstarted,therewere17possibleSierpińskinumberssmallerthan78,557,butthesehavebeeneliminatedonebyoneuntiljustsixremain:10,223,21,181,22,699,24,737,55,459,and67,607.Alongthewaytheprojecthasdiscoveredseveralverylargeprimes.

k eliminatedbyprimeoftheformk2n+1

4847 4847×23321063+15359 5359×25054502+1

(atthetimethefourthlargestknownprime)

1022319249 19249×213018586+121181226992473727653 27653×29167433+128433 28433×27830457+133661 33661×27031232+144131 44131×2995972+146157 46157×2698207+154767 54767×21337287+15545965567 65567×21013803+16760769109 69109×21157446+1

JamesJosephWho?

JamesJosephSylvester

JamesJosephSylvesterwasanEnglishmathematician,whoworkedwithArthurCayleyinmatrixtheoryandinvarianttheory,amongotherareas.Hehadalifelonginterestinpoetry,andoftenputextractsfrompoemsinhismathematicalresearcharticles.HemovedtotheUSAin1841,butreturnedshortlyafter.In1877hecrossedtheAtlanticagain,andtookupapositionasthefirstProfessorofMathematicsatJohnsHopkinsUniversity,andfoundedtheAmericanJournalofMathematics,stillgoingstrong.HereturnedtoEnglandin1883.

HisnamewasoriginallyJamesJoseph.WhenhiselderbrotheremigratedtotheUSA,theimmigrationofficialstoldhimhehadtohavethreenames:first,middle,andsurname.Forsomereasonthebrotheradded‘Sylvester’asanewsurname.SoJamesJosephdidthesame.

TheBafflehamBurglaryFromtheMemoirsofDrWatsup

LordBaffleham’sstatelyhomehadbeenburgledandsomeemeraldsandrubiesstolenfromthesafe.Soames,calledintoinvestigate,quicklybecamesuspiciousoftwovisitors,LadyEsmeraldaNickettandBaronessRubyRobham.Bothhadfallenonhardtimes,andhadnodoubtsuccumbedtotemptation.Butwherewastheproof?

Bothladiesadmittedpossessingsomejewels,butclaimedthemtobetheirown.SoameshadnotyetpersuadedInspectorRouladetoobtainasearchwarrant,whichmightsettlethematter,andhadnotbeenabletoinspecttheladies’jewelboxes.

“Thecase,”saidSoames,“hingesonjusthowmanyjewelsthetwoladiespossess.Ifthenumbersmatchwhatwasstolen,wehavethefinalpieceofevidenceweneed.Rouladeiswillingtorequestasearchwarrant,butonlyifwecantellhimthosetwonumbers.”

“Esmeraldastatedthatshehasonlyemeralds,”Imuttered,halftomyself.“AndRubysaysshehasonlyrubies.”

“Indeed.Iamsurethosestatementsaretrue.Now,thetestimonyofthemaidservantplacesthenumberofeachjewelsomewherebetween2and101,notexcludingthosetwonumbers.”

“Thecookisreluctanttotelltales,”Isaid.“ButIhavepersuadedhertotellmetheproductofthetwonumbers.”

“Andthebutler,equallyreticentbutopentopersuasionintheformoftengoldsovereigns,hastoldmetheirsum,”Soamesreplied.

“Thenwecanworkoutthetwonumbersbysolvingaquadraticequation!”Icriedinexcitement.

“Ofcourse,thoughwewon’tknowwhichnumberappliedtotheemeraldsandwhichtotherubies,”musedSoames.“Thedataaresymmetric.ButamatcheitherwaywouldbeenoughforInspectorRouladetoobtainasearchwarrant,whichwillnodoubtsuffice.”

“Ifyoutellmetheproduct,”Isaid,“Icansolvetheequation.”“Ah,mydearWatsup,youaresounsubtle,”Soamescomplained.“Letme

seeifIcandeducethenumberswithoutyourtellingme…Now,doyouknow

whatthetwonumbersare?”“No.”“Iknewthat,”saidSoames,tomyannoyance.Ifso,whyask?Butsuddenly

lightdawned.“NowIknowthenumbers,”Itoldhim.“Inthatcase,sodoI,Watsup.”

Whatwerethetwonumbers?Seepage290fortheanswer.

TheQuadrillionthDigitofπWecurrentlyknowthedecimalexpansionofπto12,100,000,000,050digits,acalculationperformedbyShigeruKondoin2013overaperiodof94days.Noonereallycareswhattheansweris,butthiskindofrecord-breakinghasledtosomeremarkablenewinsights,aswellasbeingagoodwaytotestnewsupercomputers.Oneofthemorecuriousdiscoveriesisthatitispossibletocomputespecificdigitsofπwithoutfindingthepreviousones.Butfornowwecandothisonlyinbase-16,orhexadecimal,notationfromwhichdigitsinbases8,4,and2(binary)canimmediatelybededuced.Thisideageneralisestoconstantsotherthanπ,andtobase3,butthereisnosystematictheoryyet.Nothinglikeitisknownfordecimalnotation,base10.

Theinitialdiscovery,theBBPformula,isstatedbelow:seealsoCabinetpage210.Itisaninfiniteseriesthatmakesitpossibletocalculateaspecifichexadecimaldigitofπwithoutcalculatinganyofthepreviousones.Sowecanbeconfidentthatthequadrillionthbinarydigitofπis0,thankstoProjectPiHex,andgoingevenfurtherthetwo-quadrillionthbinarydigitofπisalso0,thankstoa23-daylongcomputationbyoneofYahoo!’semployees.Despitethat,itwouldtakeanotherequallymassivecomputationtofindthepreviousdigit.

In2011DavidBailey,JonathanBorwein,AndrewMattingly,andGlennWightwickwroteasurveyofthisarea[Thecomputationofpreviouslyinaccessibledigitsofπ2andCatalan’sconstant,NoticesoftheAmericanMathematicalSociety60(2013)844–854].Theydescribedhowtocomputebase-64digitsofπ2,base-729digitsofπ2,andbase-4096digitsofanumbercalledCatalan’sconstant,startingatthe10trillionthplace.

ThestorystartswithaseriesknowntoEuler:

inΣ-notationforsums.Becauseofthepowersof2thatoccur,thisseriescanbeconvertedintoamethodforcomputingspecificbinarydigitsoflog2.Thecomputationsremainfeasible,buttakemuchlonger,asthepositionofthedigitconcernedgetslarger.

TheBBP(Bailey-Borwein-Plouffe)formulais

andthepowersof16makeitpossibletocomputespecifichexadecimaldigitsofπ.Since16=24,theseriescanalsobeusedtofindbinarydigits.

Isthisapproachlimitedtothesetwoconstants?From1997onwards,mathematicianssoughtsimilarinfiniteseriesforotherconstants,andsucceededforalargenumberofthem,including

π2log22πlog2ζ(3)π3log32π2log2π4ζ(5)

where

istheRiemannzetafunction.TheyalsosucceededforCatalan’sconstant

Someoftheseseriesyielddigitstobase3orsomepowerof3.Forexample,theamazingformulaofDavidBroadhurst

canbeusedtocomputedigitsofπ2tobase729=36.

IsπNormal?Thedigitsofπlookrandom,buttheycan’tbetrulyrandomsinceyoualwaysgetthesamenumberseverytimeyoucalculateπ,barringerrors.Itisgenerallythoughtthat,likealmostallrandomsequencesofdigits,everyfinitesequenceoccurssomewhereinthedecimalexpansionofπ.Indeed,infinitelyoften,though

withlotsofjunkinbetweensuccessiveoccurrences,andinthesameproportionyou’dexpectforarandomsequence.

Itcanbeprovedthatthisproperty,callednormality,holdsfor‘almostall’numbers:inanysufficientlylargerangeofnumbers,theproportionthatarenormalgetsascloseaswewishto100%.Butthisleavesaloophole,becauseanygivennumber,inparticularπ,mightbeanexception.Isit?Wedon’tknow.Untilrecentlythequestionlookedhopeless,butformulasliketheonesabovehaveopenedupanewlineofattack,whichmightjustsolvethequestionforbinary(orhexadecimal)digits.

Thelinkarisesthroughanothermathematicalprocedure:iteration.Herewestartwithanumber,applysomeruletogetanotherone,andrepeatedlyapplythatruletogetasequenceofnumbers.Forexample,ifwestartwith2andtheruleis‘formthesquare’thesequencegoes

241625665,6364,294,967,296…

Thebinarydigitsofanumberlikelog2canbegeneratedbytheiterativeformula

startingfromx0=0.Thesymbols(mod1)mean‘subtracttheintegerpart’,soπ(mod1)=0.14159….Thisformulawouldleadtoaproofthatlog2isnormaltobase2ifitcouldbeshownthattheresultingnumbersareuniformlyspreadovertherangefrom0to1.Such‘equidistribution’isquitecommon.Unfortunatelynooneknowshowtoproveitholdsfortheiterativeformulaabove,butit’sapromisingideaandmightgetthereeventually.

Thereisasimilarbutmorecomplicatediterativeformulaforπ:

Ifthisisequidistributed,πisnormalinbinary.Thisleadstoafinal,verystrange,discovery.Supposewestretchoutthe

rangefrom0to1byafactorof16,sothatyn=16xnrunsfrom0to16.Thentheintegerpartsofsuccessiveynrangefrom0to15.Experimentally,thesenumbersarepreciselythesuccessivehexadecimaldigitsofπ–3.Thishasbeencheckedonacomputerforthefirst10millionplaces.Ineffect,thisappearstoprovideaformulaforthenthhexadecimaldigitofπ.Thecomputationgetsharderandharderthefurtheryougo,andtook120hours.

Therearesolidreasonstoexpectthisstatementtobetrue,buttheydon’tamounttoarigorousproof.Itisknownthatveryfewerrors,ifany,occur.Sincenoneoccursforthefirst10millioniterations,thereisaboutaoneinabillionchancethatalatererrorwilloccur.However,that’snotaproof—justanexcellentreasontohopethatonecanbefound.

Afinalconjecture,alsobasedonsoundevidence,showshowstrangethisareais.Namely:nothinglikethiscanbedonefortheotherwell-knownconstante,thebaseofnaturallogarithms,roughlyequalto2.71828.Thereseemstobesomethingspecialaboutπcomparedtoe.

AMathematician,aStatistician,andanEngineer……wenttotheraces.Afterwards,theymetinthebar.Theengineerwasdrowninghissorrows.“Ican’tunderstandhowIlostallmymoney.Imeasuredthehorses,calculatedwhichwasmechanicallymostefficientandrobust,andfiguredouthowfasttheycouldrun—”

“That’sallverywell,”saidthestatistician,“butyouforgotindividualvariability.Ididastatisticalanalysisoftheirpreviousraces,andusedBayesianmethodsandmaximumlikelihoodestimatorstofindwhichhorsehadthegreatestprobabilityofwinning.”

“Anddidit?”“No.”“Letmebuyyouguysadrink,”saidthemathematician,pullingoutabulging

wallet.“Ididprettywelltoday.”Obviouslyherewasamanwhoknowssomethingabouthorses.Theothers

insistedonbeingtoldhissecret.Reluctantlyheagreed.“Consideraninfinitenumberofidenticalspherical

horses…“

LakesofWadaTopologyisoftencounterintuitive.Thismakesitdifficult,butalsointeresting.Here’sastrangetopologicalfactwithapplicationstonumericalanalysis.

Tworegionsoftheplanecanshareacommonboundarycurve;thinkoftheEnglish–ScottishborderortheAmerican–Canadianone.Threeormoreregionscanshareacommonboundarypoint:atFourCornersinAmerica,thestatesArizona,Colorado,NewMexico,andUtahallmeet.

FourCorners

Withsomeingenuityanynumberofregionscanbearrangedtosharetwocommonboundarypoints.Butitdoesn’tseempossibleforthreeormoreregionstohavemorethantwoboundarypointsincommon.Letaloneforthemalltohaveexactlythesameboundary.

However,itcanbedone.First,wehavetobepreciseaboutwhataboundarypointis.Supposewehave

someregionintheplane.Itneednotbeapolygon:itcanhaveaverycomplicatedshape—anycollectionofpointswhatsoever.Saythatapointliesintheclosureoftheregionifeverycirculardiscwithcentreatthatpointandnonzeroradius(howeversmall)containssomepointintheregion.Saythatapointliesintheinterioroftheregionifsomecirculardiscwithcentreatthatpointandnonzeroradiusiscontainedintheregion.Nowtheboundaryoftheregionconsistsofallpointsinitsclosurethatdonotlieinitsinterior.

Gotthat?Stuffontheedgebutnotinside,basically.Forapolygonalregion,boundedbyaseriesofstraightlinesegments,the

boundaryconsistsofthoselinesegments,sowhatwehavedefinedagreeswiththeusualconceptinthiscase.Itcanbeprovedthatthreeormorepolygonal

regionscannotallhavethesameboundary.Butthisisnottrueformorecomplicatedregions.In1917theJapanesemathematicianKunizōYoneyamapublishedanexampleofthreeregionsthathaveexactlythesameboundary.HesaidthathisteacherTakeoWadahadcomeupwiththeidea.Accordingly,theregions(oranythingofthesamekind)areknownastheLakesofWada.

Weconstructthethreeregionsstepbystepusinganinfiniteprocess.Beginwiththreesquareregions.

Startwiththreesquares…

Thenextendthefirstregionbyaddingatrenchthatwrapsroundallthreeregions.Dothissothateverypointontheboundaryofanysquareliesclosetothetrench.Also,makesurethatthetrenchdoesnotcloseuponitselftoleaveaholeintheresultingregion.

Digatrench…

Thenextendthesecondregionbyaddingayetnarrowertrenchthatwrapsroundallthreeregionsconstructedsofar.

Digathinnertrench…

Continuelikethat,withanevennarrowertrenchfromthethirdregion.Thengobacktothefirstregionaddingastillnarrowertrench,andsoon.

Repeatthisconstructioninfinitelyoften.Theresultingregionshaveinfinitelycomplicated,infinitelythintrenches.Butbecauseeachsuccessiveregiongetscloserandclosertoeverythingpreviouslyconstructed,allthreeregionshavethe

same(infinitelycomplicated)boundary.Thesameideaworksifwestartwithfourregionsormore:alloftheregions

constructedhavethesameboundary.TheLakesofWadawereoriginallyinventedtoshowthatthetopologyofthe

planeisnotasstraightforwardaswemightimagine.Manyyearslater,itwasdiscoveredthatsuchregionsarisenaturallyinnumericalmethodsforsolvingalgebraicequations.Thecubicequationx3=1,forexample,hasonlyonerealsolutionx=1,butitalsohastwocomplexsolutions ,wherei= .Thecomplexnumberscanbevisualisedaspointsinaplane,withx+iycorrespondingtothepointwithcoordinates(x,y).

Astandardmethodforfindingnumericalapproximationstoasolutionbeginswitharandomlychosencomplexnumber,thencalculatesasecondnumberinaspecificmanner,andrepeatsuntilthenumbersgetveryclosetogether.Theresultisthenclosetoasolution.Whichofthethreesolutionsitapproachesdependsonwhereyoustart,anditdoessoinaveryintricateway.Supposewecolourpointsinthecomplexplaneaccordingtowhichsolutiontheyleadto:saymediumgreyifthesolutionisx=1,lightgreyifthesolutionis ,darkgreyifthesolutionis .Thenthepointsthatarecolouredagivenshadeofgreydefinearegion,anditcanbeprovedthatallthreeregionshavethesameboundary.

UnlikeWada’sconstruction,theregionsherearenotconnected:theybreakupintoinfinitelymanyseparatepieces.However,itisstrikingthatregionsofsuchcomplexityarisenaturallyinsuchabasicproblemofnumericalanalysis.

Thethreeregionscorrespondingtosolutionsofthecubicequation

Fermat’sLastLimerickAchallengeformanylongagesHadbaffledthesavantsandsages.Yetatlastcamethelight:SeemsthatFermatwasright—Tothemarginaddtwohundredpages.

Malfatti’sMistakeFromtheMemoirsofDrWatsup

“Extraordinary!”Iexclaimed.Soamesglancedinmydirection,obviouslyannoyedatbeinginterruptedin

hisperusalofhisextensivecollectionofplastercastsofsquirrelfootprints.“Theanswerseemsobvious—yet,apparently,itiswrong!”Icried.“Theobvioususuallyis,”saidSoames.“Wrong,”headdedbywayof

clarification.“EverheardofGianFrancescoMalfatti?”Iasked.“Themultipleaxe-murderer?”“No,Soames,thatwas‘Hacker’FrankMacavity.”“Ah.Myapologies,Watsup,youareofcoursecorrect.Iamdistracted.My

specimenofRatufamacrouratracksisdisintegrating.Thegrizzledgiantsquirrel.”

“MalfattiwasanItaliangeometer,Soames.In1803heaskedhowtocutthreecylindricalcolumnsfromawedgeofmarbleinsuchamannerastomaximisetheirtotalvolume.Heassumedthattheproblemisequivalenttodrawingthreecirclesinsidethetriangularcrosssectionofthewedgeinsuchamannerastomaximisetheirtotalarea.”

“Anaivebutpossiblycorrectassumption,”Soamesreplied.“Thoughthecolumnsmightbecutonaslant.”

“Oh,Ihadnot…Butletussupposehisassumptiontobecorrect,sincethequestioncanalwaysbesuitablyrephrased.ItthenseemedobvioustoMalfattithatthesolutionmustcomprisethreecircles,eachtangenttotheothertwoandtoasideofthetriangle.”Idrewaquicksketch.

Malfatticircles

“Iseethefallacy,”saidSoames,inthatannoyinglyoffhandwaythatheoftenadoptswhenhehasinstantlygraspedcomplexitiesbeyondmostothermortals.

“IconfessIdonot,”Isaid.“Forifacircleliesinsidethetriangle,doesnotoverlaptheothers,andisnottangentinthatmanner,itcanbeenlarged.”

“Correct,”saidSoames.“Butthatmerelyprovesthesufficiency,notthenecessity,ofthetangencyconditions.”

“Iamawareofthat,Soames.But—howelsemightthecirclesbearranged?”“Theremightbeotherwaysforthetangencytooccur,ofcourse.For

example,Watsup:haveyouconsideredthesimplestcase,thatofanequilateraltriangle?”

Twopossiblearrangementsforanequilateraltriangle

“First,”saidSoames,“thereisMalfatti’sarrangement,theleft-handfigure.Butwhatabouttheright-handfigure?Again,nocirclecanbeenlarged,butthepatternoftangenciesisdifferent.Thesmallcirclesaretangenttothelargeone,butnottoeachother.Instead,eachistangenttotwosidesofthetriangle.”

Istaredatthefigures.“Totheeye,Soames,thefirstarrangementhaslargerarea.”

Helaughed.“Whichonlygoestoshow,Watsup,howeasilydeceivedtheeyeis.Supposethatthetrianglehassidesofunitlength.ThenMalfatti’sarrangementhasarea0.31567,buttheotheronehasarea0.31997,whichisveryslightlylarger.”

TherearetimeswhenSoames’seruditionleavesonebreathless.“Thedifferencemaybesmall,Soames,buttheimplicationisdecisive.Malfattiwaswrong.”

“Indeed.Moreover,Watsup,thedifferencebetweenMalfatti’sarrangementandthecorrectonecansometimesbemuchgreater.Forexample,ifthetriangleislong,thin,andisosceles,thenthecorrectsolutionstacksthethreecirclesontopofeachother,andtheareaisalmostdoublethatofMalfatti’sarrangement.”

Athinisoscelestriangle.Left:Malfatti’sarrangement.Right:Largestarea.

HepausedtohurlthecrumbledcastofthetracksofRatufamacrouraacrosstheroomintothefireplace.“Theironyis,”headded,“thatMalfatti’sarrangementisneverthebest.Thegreedyalgorithm—fitthelargestcirclepossibleinsidethetriangle,thenfindthelargestthatfitsintoaremaininggap,andfinallydothesameforthethirdcircle—isalwayssuperior,andindeedfindsthecorrectanswer.”

Seepage291forfurtherinformation.

SquareLeftoversPerfectsquaresendinoneofthedigits0,1,4,5,6,or9.Theydon’tendin2,3,7,or8.Infact,thefinaldigitofthesquareofanumberdependsonlyonthefinaldigitofthenumber:

Ifanumberendsin0thenitssquareendsin0.Ifanumberendsin1or9thenitssquareendsin1.Ifanumberendsin2or8thenitssquareendsin4.Ifanumberendsin5thenitssquareendsin5.Ifanumberendsin4or6thenitssquareendsin6.Ifanumberendsin3or7thenitssquareendsin9.

Numbertheoristsprefertophrasethiskindofeffectintermsofintegerstosomemodulus(page188).Ifthemodulusis10,thentheonlynumbersthatneedtobeconsideredare0–9:thepossibleremaindersondividinganynumberby10.Theirsquares(mod10)are

0149656941

andthelistofrulesforfinaldigitsofsquaresisadifferentwaytosaythesamething.

Asidefromtheinitial0,thelistofsquares(mod10)issymmetric:thenumbers1496appearafter5inreverse,6941.Thesymmetryarisesbecausenand10–nhavethesamesquaresmodulo10.Indeed,10–nisthesameas–n(mod10),andn2=(–n)2.Thesefournumbersthereforeoccurtwiceinthelist;0and5occuronce;but2,3,7,8don’toccuratall.It’snotverydemocratic,butthereyougo.

Whathappensifweuseadifferentmodulus?Thevaluesofthesquares,tothatmodulus,arecalledquadraticresidues.(Here‘residue’referstotheremainderondividingbythemodulus.)Therestarequadraticnon-residues.

Forinstance,supposethatthemodulusis11.Thenthepossibleperfectsquares(ofthenumberslessthan11)are

0149162536496481100

which,whenreduced(mod11),yield

01495335941

Sothequadraticresidues(mod11)are

013459

andthenon-residuesare

2678

Here’sashorttable:

modulusm squares(modm) quadraticresidues

2 01 013 011 014 0101 015 01441 0146 014341 01347 0142241 01248 01410141 0149 014077041 014710 0149656941 01456911 01495335941 01345912 014941014941 0149

Atfirstsight,therearefewclearpatternsasidefromthosealreadymentioned.Thisispartofthecharmofthearea,infact:whiletherearepatterns,ittakessomediggingtofindthem.Severalofthegreatestmathematiciansdevotedalotofattentiontothisarea,amongthemEulerandCarlFriedrichGauss.

Whenwesquareanumberwemultiplyitbyitself,andwhenitcomestomultiplication,whatmattermostinnumbertheoryaretheprimes.Soitpaystostartwithprimemoduli:2,3,5,7,11intheabovelist.Themodulus2isexceptional:theonlypossibleresiduesmodulo2are0and1,andbotharesquares.Forallotherprimes,abouthalfoftheresiduesaresquaresandtherestarenot.Moreprecisely,ifpisprimethenthereare(p+1)/2distinctquadratic

residuesand(p–1)/2non-residues.Thequadraticresiduesareusuallysquaresoftwodistinctnumbers,n2and(–n)2forsuitablen.However,0occursonlyoncebecause–0=0.

Compositemodulicomplicatethestory.Nowthesamequadraticresiduecansometimesbethesquareofmorethantwonumbers.Forexample1occursfourtimesformodulus8,asthesquareof1,3,5,and7.Thebestwaytomakesenseofallthisistousemodernabstractalgebra,butit’sworthtakingalookatmodulus15.Thishastwoprimefactors:15=3×5.Nowthelistofsquaresis:

n 01234567891011121314n2 01491106446101941

Sothequadraticresiduesmodulo15are

0=021=12;42;112;1424=22;72;82;1326=62;929=32;12210=52;102

Someresiduesoccuronce,sometwice,somefourtimes.Thosethatoccurfewerthanfourtimesaresquaresofnumbersthataredivisiblebyeither3or5,theprimefactorsof15.Alltheothernumbersoccurinsetsoffour,allhavingthesamesquare.

Thisisageneralpatternforanymodulusoftheformpqwherepandqaredistinctoddprimes.Thenumbersbetween0andpq–1thatarenotdivisiblebyporqsplitintosetsoffour,eachsethavingthesamesquare.(Thisfailsifoneoftheprimesis2:forexample10=2×5andwe’vealreadyseenthatinthiscasesquareseitheroccurinpairsorontheirown.)

Inalgebrawegetusedtotheideathateverypositivenumberhastwosquareroots:onepositive,theothernegative.Butinarithmeticmodulopq,mostnumbers(thosenotdivisiblebyporq)havefourdistinctsquareroots.

Thiscuriousfacthasaremarkableapplication,towhichwenowturn.

CoinTossingoverthePhoneSupposethatAliceandBobwanttoplayacoin-tossinggamewitha50–50outcome.Aswe’vealreadyseen(page131)AliceisinAliceSpringsandBobisinBobbington.Cantheytossthecoinoverthephone?Thebigsnagisthesameasforpoker.IfAlicetossesacoin,or,equivalently,carriesoutanyactivitywithtwoequallyprobableresults,andtellstheresulttoBob,hehasnoideawhethersheistellingthetruth.NowadaystheycoulddoitoverSkypeandwatchthecoinbeingtossed,buteventhen,thetosscouldberiggedbyfilmingseveraltossesaheadoftimeandsendingoneofthoseinstead.

Tossingacoinislikeplayingpokerwithapackofjusttwocards,sotheycouldadaptthemethodonpage132.However,there’sanotherelegantwaytoachievethesameresult,usingquadraticresidues.Here’show.

Alicechoosestwolargeoddprimenumberspandq.Shekeepsthemprivate,butsendstheirproductn=pqtoBob.YoumightthinkthatBobcouldthenfindpandqbyfactorisingn,butasfarasiscurrentlyknown,thereisnopracticalmethodtodothatwhenthenumbersaresufficientlylarge—say100digitsforeachofpandq.Thefastestcomputerusingthefastestknownalgorithmwouldtakelongerthanthelifetimeoftheuniverse.SoBobmustremainignorantoftheactualprimesinvolved.However,thereareveryquickwaystotesta100-digitnumbertoseewhetheritisprime.SoAlicecanfindpandqbytrial-and-error.

Bobchoosesarandomintegerx(modn),whichhekeepsprivate.Ifheisextremelypedantic,hecanquicklycheckwhetherxisamultipleofp

andq:notbydividingbythosenumbers,sincehedoesn’tknowthem,butbyfindingthehcfofxandnusingEuclid’salgorithm(page110).Iftheresultisnot1hethenknowseitherporq,sotheprocesshastoberepeatedwithanewx.Butinpracticeheneednotbother,sincewhenpandqhave100digitstheprobabilitythatporqdividesarandomlychosenxis2×10–100.

Bobnowcalculatesx2(modn),whichcanalsobedonequickly,andsendsittoAlice.TheyhaveagreedthatifAlicecancorrectlydeduceeitherxor–xshewins(‘heads’).Otherwise,sheloses(‘tails’).

BythepreviousitemAliceknowsthatintegersmodulopqthatarenotdivisiblebyporqhaveexactlyfoursquareroots.Sincexand–xhavethesamesquare,theseareoftheforma,–a,b,–bforsuitableaandb.Aliceknowsp,q,andx,whichimpliesthatshecancomputethesefoursquarerootsquickly.Two

ofthemmustbeBob’sxand–x;theothertwomustbedifferent.SoAlicehasa50%chanceofguessing xcorrectly—equivalenttotossingafaircoin.Shechoosesoneofthesefour,sayb,andsendsittoBob.

BobtellsAlicewhetherb= xornot;thatis,whethersheisrightorwrong.Ah—buthowdowestopBobcheating?AndhowdoesBobknowthatAlice

hasdonewhatsheissupposedtodo?Whetherornotb= x,BobcanbehappythatAlicehasplayedfairlyby

computingb2(modn).Thisshouldbethesameasx2.IfAliceloses,shecanconvinceherselfthatBobhasnotliedbyaskinghim

tosendhertheprimefactorspandqofn.Normallythiswouldbeimpossible,butifAlicehaslost,thenBobknowsallfoursquarerootsofx2,andthereisanumber-theoretictricktocalculatepandqquicklyfromthisinformation.Infact,thehighestcommonfactorofaandbisoneofthetwoprimes,andagainitcanbefoundusingEuclid’salgorithm.Theothercanthenbefoundbydivision.

HowtoStopUnwantedEchoesQuadraticresiduesmayseemtypicaloftheabstruseexplorationsofpuremathematicians:anintellectualgamewithnopracticaluses.Butit’samistaketothinkthatamathematicalideaisuselessjustbecauseitdoesn’tobviouslyderivefromapracticalproblemineverydaylife.It’salsoamistaketothinkthateverydaylifeisasstraightforwardasitappearstobeonthesurface.Evensomethingassimpleasapotofjaminasupermarketinvolvesmakingtheglass,growingthesugarcaneorbeet,refiningthesugar…andprettysoonyou’reintostatisticalhypothesistestingfordisease-resistantfruitandthedesignoftheshipusedtotransportvariouscomponentsorthefinishedproductroundtheglobe.Inaworldof7billionpeople,massfoodproductionisnotjustamatterofpickingsomeblackberriesandboilingthemup.

It’struethatthemathematicianswhofirstcameupwiththeseideashadnoparticularapplicationsinmind;theyjustthoughtquadraticresidueswereinteresting.Buttheywerealsoconvincedthatunderstandingthemwouldbeapowerfuladditiontothemathematicaltoolkit.Practicalpeoplecan’tuseatoolunlessitexists.Andwhileitmightseemtomakesensetowaitforanapplicationbeforeinventingasuitabletool,we’dstillbesittingincavesifwe’ddoneitthatway.“Whyareyouwastingtimebashingthoserockstogether,Ug?Youshouldbebashingmammothsovertheheadwithastickliketheotherboys.”

Quadraticresidueshavemanydifferentuses.Oneofmyfavouritesisthedesignofconcerthalls.Whenmusicreflectsoffaflatceiling,theresultisadistinctecho,whichdistortsthesoundandisgenerallyunpleasant.Ontheotherhand,aceilingthatabsorbssoundmakestheperformancesounddeadandfuzzy.Togetgoodacoustics,thesoundhastobeallowedtobounceback,butasadiffusespreadofsoundsratherthanasharpecho.Soarchitectsfitdiffusersontheceiling.Thequestionis:whatshapeshouldthediffusersbe?

Quadraticresiduediffuser(mod11)

In1975ManfredSchroederinventedadiffuserconsistingofaseriesofparallelgrooves,whosedepthsarederivedfromthesequenceofquadraticresiduesforsomeprimemodulus.Forexample,supposethattheprimeis11.We’vejustseenthatthesquaresof0–10,reducedmodulo11,are:

01495335941

andthesequencerepeatsthesevaluesperiodicallyforlargernumbers.It’ssymmetricaboutthemiddle,betweenthetwo3’s,becausex2=(–x)2moduloanyprime.Comparethepicturebelow,showingthesenumbersasrectangles,withtheshapeofthediffuserabove.Noticethatinthiscasethedepthsofthegroovesareobtainedbysubtractingtheresiduesfromsomeconstantdepth.Thishasnoseriouseffectonthemainmathematicalpoint.

Graphofquadraticresidues(mod11)

What’ssospecialaboutquadraticresidues?Onefeatureofasoundwaveisitsfrequency:howmanywaveshittheeareverysecond.Highfrequenciesgivehighnotes,lowfrequenciesgivelownotes.Arelatedfeatureisthewavelength:thedistancebetweensuccessivepeaks.High-frequencywaveshaveshortwavelength,andlow-frequencywaveshavelongerwavelength.Wavesofagivenwavelengthtendtoresonatewithcavitiesinthesurfacewhosesizeissimilartothatwavelength.Sowaveswithdifferentfrequenciesreactdifferentlywhentheyhitasurface.

Thequadraticresiduediffuserhasadelightfulmathematicalproperty:waves

withmanydifferentfrequenciesreacttoitinthesamemanner.Technically,itsFouriertransformisconstantacrossarangeoffrequencies.Schroederpointedoutanimportantconsequence:thisshapediffusessoundwavesofmanydifferentfrequenciesinthesamemanner.Inpracticethewidthsofthegroovesarechosentoavoidtherangeofwavelengthsthathumanscanhear,andtheirdepthsareaspecificmultipleofthesequenceofquadraticresidues,relatedtothewidth.

Whenthegroovesareparallel,asinthepicture,thesoundisdiffusedsideways,atrightanglestothedirectionofthegrooves.Thereisatwo-dimensionalanalogue,asquarearrayofrodsalsobasedonquadraticresidues,andthisdiffusesthesoundequallyinalldirections.Diffuserslikethatareoftenfoundinrecordingstudios,toimprovethesoundbalanceandgetridofextraneousnoises.

SoalthoughEulerandGausshadnoideawhattheirinventionwouldbeusedfor,orindeedwhetheritwouldeverbeusedforanything,itoftenplaysacrucialrole,behindthescenes,whenyoulistentorecordedmusic—beitclassical,jazz,country,rock,hiphop,crossoverthrashmetal,orwhateverelsetakesyourfancy.

Seepage292forfurtherinformation.

TheEnigmaoftheVersatileTileFromtheMemoirsofDrWatsup

“Solvingacrimeisoftenlikenedtofittingtogetherthepiecesofajigsawpuzzle,”Soamesremarkedoutoftheblue.Thebluebeingacloudofsmokethatenvelopedhishead,emanatingfromhispipe.

“Anaptsimile!”saidI,raisingmyheadfrommynewspaper.Hesmiledslyly.“Notso,Watsup.Onthecontrary,averypoorone.When

investigatingacrime,wedonotknowwhatthepiecesmightbe,norwhetherweareinpossessionofthemall.Notknowingthepuzzle,howcanwebecertainoftheanswer?”

“Surely,Soames,thatbecomesevidentwhenenoughoftheknownpiecesfittogetherintoanelegantpattern.”

Hesighed.“Ah,buttherecanbesomanypieces,Watsup.Andsomanypatterns.Decidingwhichiscorrecttakesacertain…jenesaisquoi.ButIdon’tknowwhat.”

Atthatmomenttherewasaknockonthedoorandawomanrushedin.“Beatrix!”Icried.“Oh,John!Ithasbeenstolen!”Andsherushedintomyarms,sobbing.Idid

mybesttocomforther,thoughIconfessmyownheartwasracing.Afteratime,shebecamecalm.“Pleasehelpme,MrSoames!Itisaruby

pendantinheritedfrommylatemother.Ilookedforitthismorning,andithadgone!”

“Donotdistressyourself,mydear,”saidI,pattinghershoulder.“SoamesandIwillapprehendthethiefandrecoverthejewel.”

“Didyoucomebycab?”Soamesasked.“Yes.Itiswaitingoutside.”“Thenweshalllosenotimeininvestigatingthesceneofthecrime.”Afterhalfanhourcrawlingonthefloor,samplingdustfromthecornersof

severalrooms,andinspectingthedoorstepandflowerbeds,Soamesshookhishead.“Nosignofabreak-in,MissSheepshear.Buttherearesmallscratchmarksonyourjewelcase.Veryfresh,andnotyours,forwhoevermadethemisleft-handed.”Heputthecasedown.“Haveanystrangersvisitedthehouserecently?Tradesmen,perhaps?”

“No…Oh!Thetilers!”

Twomenclaimingtobetilershadcometothebackdoorofferingtorenovatethebathroom.“Itisthenewfashion,MrSoames.Plainwhitesquaretiles,intowhichhasbeencutabluemotifmadefromtilesofamoreelaborateshape.TheDimworthyshadtheirsdonelastmonth,andfather—”Hervoicefailed,overcome,closetotears.Itookherhand.

“Areyouinthehabitofengagingunknowntradesmenatthedoorstep?”Soamesenquired.

“Why,no,MrSoames.Ordinarilywewoulddealonlywithareputablefirm.Buttheyareallbookedsolidformonths.Andtheseseemedhonest,decentmen.”

“Theyalwaysdo.Didyouleaveeitherofthemunattended?”Shethoughtforamoment.“Yes.Theassistantwaslefttomake

measurementsinthebathroomwhilehismastershowedmesamplemotifs.”“Ampletimetostealasmallbutvaluableitem.Theyareclever:bynotbeing

greedytheyensuredthatthetheftwasnotimmediatelynoticed.Didtheyleaveanydocumentation?”

“No.”“Havetheyreturnedsince?”“No,Iamawaitingawrittenestimateforthework.”“Iventuretopredictthatitwillnotarrive,madam.Itisamodusoperandi

thatinthetradewecalla‘distractionburglary’.”Overthefollowingweek,asteadystreamofladiesengagedSoames’s

serviceswithsimilartales.Thetradesmenvariedinappearance,butSoameswasunsurprised.“Disguised,”hesaid.

Thebreakthroughcameonthethirteenthcase,atthehomeofMrsAmeliaFotherwell.Soamesnoticedalumpofmudadheringtothebathroomdoor,inwhichwasembeddedasmallpieceofbone.ThecompositionofthemudandthenatureoftheboneledtoadirtybackyardnexttoasardinecanneryinoneofamazeoftinystreetsbehindtheAlbertDock.

“Sonowwebreakinandsearchforevidence?”saidI,reachingformyrevolver.

“No:thatmightalertthethief.WereturntoBakerStreetandassembleourcase.”

“Tellme,Watsup,”hesaid,aswesharedabottleofport.“Whatfeaturesdoallthesetheftshaveincommon?”Ipointedoutthosethatsprangtomind.“Verygood.Butyouhaveomittedthemostsignificantfeature.Themotifs.Youhavealistofthem,nodoubt?”

Itookoutmynotebook.Itread:

MrsWotton:threetilesforminganequilateraltriangle.Beatrix:fourtilesformingasquare.MissMakepiece:fourtilesformingasquarewithasquarehole.TheCranfordtwins:fourtilesformingarectanglewitharectangularhole.MrsBroadside:fourtilesformingaconvexhexagon.MrsProbert:fourtilesformingaconvexpentagon.LadyCunningham:fourtilesforminganisoscelestrapezium.MissWilberforce:fourtilesformingaparallelogram.MrsMcAndrew:fourtilesformingwindmillsails.MrsTushingham:sixtilesformingahexagonwithahexagonalhole.MissBrown:sixtilesforminganequilateraltrianglewithtrianglescutoffthecorners.DameJenkin-Glazeworthy:twelvetilesformingadodecagon(regular12-sidedpolygon)witharegular12-sidedstarhole.MrsFotherwell:twelvetilesformingadodecagonwith12-sidedstarholeshapedlikethebladeofacircularsaw.

“Aremarkablecollection,”saidSoames.“IthinkitistimetosendaBakerStreetIrreducibletoInspectorRoulade,askinghimtoraidthepremisesnearAlbertDock.”

“Whatareyouexpectingthepolicetofind?”“Recall,Watsup:eachladytoldusthathermotifwasmadefromanumber

ofidenticaltiles.”“Yes.”“Butthemotifsareveryvaried,suggestingthatalthougheachmotifuseda

singleshape,differentmotifsrequireddifferentshapesoftile.Theladiescannotdescribetheshapeexceptas‘irregular’,sowehavenoevidencethatthesameshapewasemployedforeachmotif.Ithereforeexpectthepolicetofindthirteenboxesofstrangelyshapedtiles:oneforeachmotif.”

AfteracoupleofhoursMrsSoapsudsappeared.“InspectorRoulade,MrSoames.”

TheInspectorentered,accompaniedbyaconstablebearingabox.“Ihaveplacedtwosuspectsunderarrest,”saidtheInspector.

“Roland‘therat’Ratzenbergand‘Bugface’McGinty.”“Yes,buthowonEarth—oh,nevermind.Icanholdthemfortwenty-four

hours.Buttheevidenceisweak.”Soameslooksshocked.“Surelyyoufoundallthoseboxesoftiles?Isnotthat

onemerelyasample?”TheInspectorshookhishead.“No:itistheirentirety.”

Soameswalkedovertotheboxandopenedit.Itcontainedtwelveidenticaltiles.“Oh,”saidhe.

“Itseemsthecasehascollapsed,”Iventured.“Icannotbelievethatsuchavariedlistofmotifscanallbemadefromasingleshapeoftile.”

ButSoamessuddenlybecamemoreanimated.“Youmaywellberight,”hesaid.“Unless…”Heproducedrulerandprotractorandbeganmeasuringoneofthetiles.

Afterafewmoments,asmilestoleacrosshisfeatures.“Clever!”saidhe.“Veryclever.”Heturnedtowardsme.“Ihavebeenextremelyfoolish,Watsup,andassumedwhenIshouldhavemaintainedanopenmind.DoyourememberourtopicofconversationjustbeforeBeatrixarrivedindistress?”

“Um—jigsawpuzzles.”“Indeed.Andthiscasehingesupononeofthemostremarkablejigsaw

puzzlesIhaveeverencountered.Lookatthistile.”“Itseemsaveryordinaryquadrilateral,”Isaid.“No,Watsup:itisaveryextraordinaryquadrilateral.Allowmeto

demonstrate.”Andhedrewadiagram.

Theversa-tile(thedashedlineisincludedtoexplainthegeometry)

“ThesidesABandBCareequal,andABCisarightangle,soanglesBACandBCAare45°,”Soamesexplained.“AngleACDis15°,soBCDis60°.AngleADCisagainarightangle,andthatmakesangleCADequalto75°.”

TheInspectorandIremainedunenlightened.Soameshandedmefourtiles.“Watsup,tryfittingthesetogethertomakeanelegantshape.Muchasadetectivemightfitcluestogethertocreateanelegantdeduction,toquoteyourearlieranalogy.”

“MayIturnthemover?”

“Anexcellentquestion!Yes,youmayturnanypieceoverifyouwish.”Iexperimentedforawhile.Suddenly,theanswerappearedbeforeme.

“Soames!Theymakeasquare—Beatrix’smotif!Howbeautiful!”

Watsup’sarrangement

Soamespeeredatmylittlejigsaw.“Indeeditis.Doyoustillcontendthatanelegantexplanationofhowseveralcluesfittogetherconstitutesdefinitiveevidencethatyouhavefoundtheguiltyparty?”

“Howelsecouldtheevidencefitsotightly,Soames?”“Howelseindeed?”Irealisedthathisquestionwasrhetorical.“Thereisa

holeinyourargument,Watsup,”hewenton,whenIdeclinedtoanswer.“Letuseliminateit.”Hereacheddownandrearrangedthepiecestoformacompletesquare.

Soames’salternative

“Oh,”saidI,shamefaced.“ThatisBeatrix’smotif,then.”“SoIconjecture.Butbenotdowncast:yourarrangementisMiss

Makepiece’s.”Lightdawned.“Youthinkthatcopiesofthisonetilecanformallthirteen

motifs?”

“Iamcertainofit.See:hereishowthreetilesformMrsWotton’smotif,anequilateraltrianglewithatriangularhole.”

Thethirdarrangement

“Goodheavens,Soames!”“Itisaremarkablyversatile—er—tile,”hereplied.“Thankstoitscunning

geometry.”“Soallwehavetodo—”Ibegan.“—istofindarrangementsthatfittheothertenmotifs!”Rouladefinishedfor

me.Soamesbegantoreamouthispipe.“IamsureIcansafelyleavethattoyou

gentlemen.”Thatevening,ItookacabtoBeatrix’sfather’shouse,stoppingonlytopick

upsomethingfromthejeweller’s.Shereceivedmeinthedrawing-room.Iplacedalongboxonthetable.“Openit,mydear.”Shereachedforithesitantly,hopedawningonherlovelyface.“Oh!John,youhaverecoveredmypendant!”Shetookmyhand.“HowcanI

everthankyou?”Suddenlyshefellsilent.“But—thisisnotmine.”Shereachedintotheboxandpickedoutasparklingjewel.“Itisanengagementring.”

“Soitis.Anditcanbeyours,”saidI,goingdownononeknee.

Canyoufindtheothertenarrangements?Seepage292fortheanswers.

TheThrackleConjectureAgraphisacollectionofdots(nodes)joinedbylines(edges).Whenagraphisdrawnintheplane,theedgesoftencross.In1972JohnConwaydefinedathrackletobeagraphdrawnintheplaneforwhichanytwoedgesmeetatanodeandotherwisedonotcross,ordonotmeetatanodebutcrossexactlyonce.ThenameissaidtohavebeeninspiredbyaScottishfishermancomplainingthathislinewas‘thrackled’—tangled.

Twothrackles

Thefigureshowstwothrackles.Theleft-handonehas5nodesand5edges,whiletherighthandonehas6nodesand6edges.Conwayconjecturedthat,foranythrackle,thenumberofedgesislessthanorequaltothenumberofnodes.Heofferedaprizeofabottleofbeerforaproofordisproof,butastheyearspassedandnoonesolvedtheriddle,theprizeroseto$1,000.

Boththracklesshownareclosedloops(nodesonacircle),drawnwithoverlaps.Itisknownthatanyclosedloopwithn≥5nodescanbedrawntoformathrackle.Ifso,thenumberEofedgescanbeequaltothenumbernofnodeswhenevern≥5.PaulErdoősprovedthattheconjectureistrueforanygraphwithstraightedges.ThebestboundonthesizeofEwasprovedbyRadoslavFulekandJánosPachin2011:

ForfurtherinformationSeepage292.

BargainwiththeDevilAmathematician,whohasspenttenfruitlessyearstryingtoprovetheRiemannHypothesis,decidestosellhissoultotheDevilinexchangeforaproof.TheDevilpromisestoshowhimtheproofinaweek,butnothinghappens.

Ayearlater,theDevilturnsupagain,lookinggloomy.“Sorry,couldn’tproveiteither,”hesays,handingbackthemathematician’ssoul.Hepauses,andhisfacelightsup.“ButIthinkIfoundareallyinterestinglemma…“

Atriskofspoilingthejoke,I’dbetterexplainthatinmathematicsalemmaisaminorpropositionwhosemaininterestistobeapotentialsteppingstonetowardssomethinginterestingenoughtodeservebeingcalledatheorem.Thereisnologicaldifferencebetweenatheoremandalemma,butpsychologicallytheword‘lemma’indicatesthatwhathasbeenprovedgoesonlypartwaytowardswhatisreallyrequired—

I’llgetmecoat.

ATilingThatIsNotPeriodicManydifferentshapestiletheplanewithoutleavinggapsoroverlapping.Theonlyregularpolygonsthatcandothisaretheequilateraltriangle,thesquare,andthehexagon.

Thethreeregularpolygonsthatcantiletheplane

Ahugerangeoflessregularshapesalsotiletheplane,suchastheseven-sidedpolygoninthenextpicture.Itisobtainedfromaregularheptagonbyflippingthreeofitssidesoveracrossthelinejoiningtheirends.

Left:Howtomaketheseven-sidedpolygontilefromaregularheptagon.Right:Spiraltiling.

Theregularpolygontilingsareperiodic,thatis,theyrepeatindefinitelyintwodifferentdirections,likewallpaperpatterns.Thespiraltilingisnotperiodic.However,theseven-sidedpolygonthatoccurstherecanalsotiletheplaneperiodically.

Howcanthisbedone?Seepage293fortheanswer.

Aretheretilesthatcantiletheplane,butcannotdosoperiodically?Thisquestionhasdeepconnectionswith

mathematicallogic.In1931KurtGödelprovedthatthereexistundecidableproblemsinarithmetic:statementsforwhichnoalgorithmcandecidewhethertheyaretrueorfalse.(Analgorithmisasystematicprocessthatisguaranteedtostopwiththecorrectanswer.)Histheoremimpliesanother,moredramaticone:therearestatementsinarithmeticthatcanneitherbeprovednordisproved.

Hisexampleofsuchastatementwasrathercontrived,andlogicianswonderedwhethermorenaturalproblemsmightalsobeundecidable.In1961HaoWangwasthinkingaboutthedominoproblem:givenafinitenumberofshapesfortiles,isthereanalgorithmthatcandecidewhetherornottheycantiletheplane?Heshowedthatifthereexistsasetoftilesthatcantiletheplane,butcannottileitperiodically,thereisnosuchalgorithm.HisideawastoencoderulesoflogicintotheshapesofthetilesanduseresultslikeGödel’s.Itworked,too:in1966RobertBergerfoundasetof20,426suchtiles,provingthatthedominoproblemisindeedundecidable.

Twentythousanddifferentshapesisratheralot.Bergerreducedthenumberto104;thenHansLa¨uchligotitdownto40.RaphaelRobinsonreducedthenumberofshapestosix.RogerPenrose’sdiscoveryin1973ofwhatwenowcallPenrosetiles(seeCabinetpage116)reducedthenumberstillfurther,toameretwo.Thisleftanintriguingmathematicalmystery:isthereasingleshapeoftilethatcantiletheplane,butcannottileitperiodically?(Thetile’smirrorimagecanbeusedaswell.)Theanswerwasfoundin2010byJoshuaSocolarandJoanTaylor[Anaperiodichexagonaltile,JournalofCombinatorialTheorySeriesA118(2011)2207–2231],anditis‘yes’.

Theirtileisshowninthefigure.Itisa‘decoratedhexagon’,withextra‘matchingrules’,anditdiffersfromitsmirrorimage.Thedecorationshavetofittogetherasshown.

FourcopiesoftheSocolar–Taylortileillustratingmatchingrules

Thenextfigureshowsthecentralregionofatilingoftheplane.Youcanseethatitdoesn’tlookperiodic.Thepaperexplainswhythistilingcanbecontinuedtocovertheentireplane,andwhytheresultcannotbeperiodic.Seetheirpaperfordetails.

CentralregionofatilingoftheplanebySocolar–Taylortiles

TheTwoColourTheoremFromtheMemoirsofDrWatsup

“Well,Soames,hereisajollylittleconundrumtobrightenyourmood.”ItossedtheDailyReporteracrosstomyfriendandcompanion,thenearlyfamousdetective,whowascurrentlysufferingfromafitofdepressionbecausehiscompetitoracrossthestreetwasdecidedlymorefamous,andlikelytoremainthatway.

Hetosseditbackwithasneer.“Watsup,Ihaveinsufficientenergytoread.”“ThenIshallreadittoyou,”Ireplied.“Itseemsthatthecelebrated

mathematicianArthurCayleyhaspublishedanarticleintheProceedingsoftheRoyalGeographicalSociety,askingwhether—”

“Whetheramapmaybecolouredwithatmostfourcolourssothatadjacentregionsreceivedifferentcolours,”Soamesinterrupted.“Itisalong-standingproblem,Watsup,andIfearitwillnotbeansweredinourlifetimes.”Isaidnothing,hopingtodrawhimout,sincethiswasthelongestsentencehehadutteredforthebestpartofaweek.Myployworked,forafteranawkwardsilencehecontinued:“AyoungmannamedFrancisGuthrieposedtheproblemtwoyearsbeforeIwasborn.Unabletosolveit,heemployedthegoodofficesofhisbrotherFrederick,astudentofProfessorAugustusDeMorgan.”

“Ah,yes,Gussie,”Iinterjected,havinghadsomeacquaintancewiththefamilyofthisadmirableeccentric,authorofABudgetofParadoxesandscourgeofmathematicalcrackpotseverywhere.

“DeMorgan,”Soameswenton,“madenoprogress,soheaskedthegreatIrishmathematicianSirWilliamRowanHamilton,whogavehimshortshrift.AndtheretheproblemlanguisheduntilCayleyonceagaintookitup.Thoughwhyhechosetopublishinthatparticularjournal,Ihavenoidea.”

“Possibly,”Iessayed,“becausegeographersareinterestedinmaps?”ButSoameswashavingnoneofit.

“Notinthismanner,”hehuffed.“Ageographerwillcolourregionsofamapaccordingtopoliticalconsiderations.Adjacencywouldnotaffectthechoice.Why,Kenya,Uganda,andTanganyikaarealladjacent,butonanymapoftheBritishEmpireallthreewillbecolouredpink.”

Iadmittedthatthiswasso.OurdearQueenwouldnotbeamusedwereit

otherwise.“ButSoames,”Ipersisted,“itremainsanintriguingquestion.Allthemoresobecausenooneseemsabletoanswerit.”

Soamesgrunted.“Letusmaketheattempt,”saidI,andquicklydrewamap.“Curious,”saidSoames.“Whyhaveyoumadeeveryregioncircular?”“Becauseanyregionwithoutholesistopologicallyequivalenttoacircle.”

Watsup’smap,andhowtocolourit

Soamespursedhislips.“Evenso,itisapoorchoice,Watsup.”“Why?Itseemstome—”“Watsup,manythingsseemtoyou,butfewactuallyare.Althoughanysingle

regionistopologicallyacircle,twoormoreregionsmayoverlapinamannerthatisimpossiblefortwoormorecircles.Asevidence,yourmaprequiresonlytwocolours.”Heshadedinabouthalfoftheregions.

“Well,yes,butIamcertainthatamoreelaboratemapofthesamekind—”Soamesshookhishead.“No,no,Watsup.Anymapcomposedentirelyof

circularregions,eveniftheybeofdifferentsizes,andoverlapinacomplicatedmanner,canbecolouredwithtwocolours.Assuming,asisalwaysthecaseinsuchquestions,that‘adjacent’requiresregionstoshareacommonlengthofborder,notjustisolatedpoints.”

Myjawdropped.“Atwocolourtheorem!Astonishing!”Soameshadthegracetoshrug.“Buthowcansuchatheorembeproved?”

Soamesleanedbackinhischair.“Youknowmymethods.”

Seepage293fortheanswer.

TheFourColourTheoreminSpaceSoameswasreferringtothecelebratedFourColourTheorem,whichtellsusthatgivenanymapintheplane,itsregionscanbecolouredusingatmostfourdistinctcolours,sothatregionsthatshareacommonborderhavedifferentcolours.(Here‘shareaborder’requiresthecommonbordertobeofnonzerolength;thatis,meetingatasinglepointdoesn’tcount.)Thisresultwasconjecturedin1852byFrancisGuthrie,andprovedin1976byKennethAppelandWolfgangHakenusingmassivecomputerassistance.*Theirproofhassincebeensimplified,butacomputerisstillessentialtocarryoutalargenumberofroutinebutcomplicatedcalculations.

Mighttherebeananalogoustheoremfor‘maps’inspaceratherthanonaplane?Nowtheregionswouldbesolidblobs.Alittlethoughtshowsthatsuchamapcanrequireanynumberofcolours.Suppose,forinstance,thatyouwantamapneedingsixcolours.Startwithsixdistinctspheres.Letsphere1putoutfivethintentaclesthattouchspheres2,3,4,5,and6.Thenletsphere2putoutfourthintentaclesthattouchspheres3,4,5,and6.Nowmoveontosphere3,andsoon.Theneachtentacledregiontouchesallfiveothers,sotheymustallhavedifferentcolours.Ifyoudidthiswith100spheres,you’dneed100colours;withamillionspheres,you’dneedamillioncolours.Inshort,thereisnolimittothenumberofcoloursrequired.

A‘map’inspaceneedingsixcolours

In2013BhaskarBagchiandBasudebDatta[Higher-dimensionalanalogues

ofthemapcoloringproblem,AmericanMathematicalMonthly120(October2013)733–736]realisedthatthisisnottheendofthestory.Thinkof‘maps’formedbyafinitenumberofcirculardiscsintheplanethateitherdonotoverlaportouchatacommonpoint.Supposeyouwanttocolourthediscssothattouchingdiscsgetdifferentcolours.Howmanydoyouneed?Itturnsoutthatagaintheansweris‘atmostfour’.

Infact,thisproblemisessentiallyequivalenttotheFourColourTheorem.Thistheoremcanbereformulatedastheproblemofcolouringthenodesofanetwork(orgraph,Seepage201)intheplane,withnoedgescrossingeachother,sothatiftwonodesarejoinedbyanedgethenthenodesreceivedifferentcolours.Justcreateanodeforeachregionofthemapandanedgebetweentwonodesifthecorrespondingregionsshareacommonboundary.Itcanbeprovedthatanynetworkintheplanecanbeproducedfromasuitablesetofcirclesbyjoiningthecentresofthosecirclesthattouch.Forinstance,hereisasetofcirclesneedingfourcolours,theassociatednetwork,andamapwithatopologicallyequivalentdistortionofthatnetworkthatalsoneedsfourcolours.

Left:Fourcirclesandtheassociatednetwork(greydotsandlines).Right:Amapwithatopologicallyequivalentnetwork,needingfourcolours.

Wecanextendthediscformulationnaturallytothreedimensions,byusingspheresinsteadofdiscs.Againeitherthespheresdonotoverlaportheytouchatacommonpoint.Supposeyouwanttocolourthespheressothattouchingspheresgetdifferentcolours.Howmanydoyouneed?BagchiandDattaexplainedwhythisnumbermustbeatleast5andnogreaterthan13.Theexactnumberiscurrentlyamathematicalmystery.Butyoumaybeabletoprovethatatleastfivecoloursareneeded.Theirresultimpliesthatsomethree-dimensionalmapsarenotequivalenttothoseobtainedfromspheres.

Seepage295fortheanswer.

Footnote

*SeeCabinet(pages10–16)foradiscussionoftheproblemanditseventualsolution.

ComicalCalculusForthisoneyouneedtoknowsomecalculus.If integrationsymbol,thentheexponentialfunctionexisitsownintegral:

ex= ex

Therefore

(1– )ex=0

So

Thecalculationseemstobenonsense;eventhefirstlinereallyoughttobeex= exdx.Andalatersteptakestheformula

1+y+y2+y3+y4+…=(1–y)–1

forsumminganinfinitegeometricseriesandreplacesyby .Thisformulaisvalidwhenyisanumber,lessthan1.But isn’tevenanumber,justasymbol.Howludicrous!

Despitewhich,thefinalresultisthecorrectpowerseriesforex.Thisisn’tacoincidence.Withtherightdefinitions(forexample, isan

operator,transformingafunctionintoitsintegral,andthe‘geometricseries’formulaworksforoperatorsundersuitabletechnicalconditions)everythingcanbemadeperfectlylogical.Butitdoeslookstrange.

TheErdoősDiscrepancyProblem

PaulErdoős

PaulErdoőswasaneccentricbutbrilliantHungarianmathematician.Heneverownedahouseorhadaregularacademicjob,preferringtotraveltheworldwithasuitcaseandsleepinthehomesofunderstandingcolleagues.Hepublished1,525mathematicalresearchpapers,collaboratingwith511othermathematicians—afigurenooneelsehascomenear.Hepreferredingenuitytodeepsystematictheories,andhisdelightwastosolveproblemsthatlookedsimple,butturnedoutnottobe.Hismainachievementswereincombinatorics,buthecouldturnhishandtomanyareasofmathematics.HefoundamuchsimplerproofofBertrand’spostulate(thereisalwaysaprimebetweennand2n)thantheoriginalanalyticoneofPafnutyChebyshev.Ahighpointofhiscareerwasaproofoftheprimenumbertheorem(thenumberofprimeslessthanxisapproximatelyx/logx)thatavoidedcomplexanalysis,previouslytheonlyknownroutetoaproof.

Hehadalifelonghabitofofferingmonetaryprizesforsolutionstoproblemsthathecameupwithbutwasunabletosolvehimself.Hewouldoffer$25forasolutionofsomethinghesuspectedtoberelativelyeasy,andthousandsofdollarsforsomethinghebelievedwasreallyhard.AtypicalexampleofhiskindofmathematicsistheErdoősDiscrepancyProblem,pricedat$500.Itwasposedin1932andsolvedearlyin2014.Itisaremarkableexampleofhowtoday’smathematiciansapproachlong-standingmysteries.

Theproblemstartswithaninfinitesequenceofnumbers,eacheither+1or–

1.Itmightbearegularsequence,suchas

+1–1+1–1+1–1+1–1+1–1…

oranirregular(‘random’)onesuchas

+1+1–1–1+1–1+1+1–1+1…

whichIgotbytossingacoin.Itneednotcontainthesameproportionof=and=signs.Anysequencewilldo.

Onewaytoseethatthefirstoftheseisregularistolookateverysecondterm:

–1–1–1–1–1…

Thesumsofthefirstntermsgo

–1–2–3–4–5…

anddecreasewithoutlimit.Ifwedothesameforthesecondsequenceweget

+1+1–1+1–1…

withsums

+10–10+1…

thatgoupanddown.Supposewetakeaspecificbutarbitrarysequenceof 1’s,andsetourselves

apositivetargetnumberC,whichcanbeaslargeaswewish—abillion,say.Erdoősaskedwhetherthereisalwayssomenumberdsuchthatthesumsoftermsdstepsapart,atpositionsd,2d,3d,andsoon,eitherbecomelargerthanCorsmallerthan-Catsomestage.

Havingreachedthetarget,theymaysubsequentlygivesumsbetweenCand-C:itisenoughtohitthetargetonce.However,therehastobeasuitablestepsizedforanytargetC.Ofcourse,ddependsonC.Thatis,ifthesequenceisx1,x2,x3,…,canwefinddandksothat

|xd+x2d+x3d+…+xkd|>C?

Theabsolutevalueofthesumontheleftisthediscrepancyofthesub-sequencedeterminedbythestep-sized,anditmeasurestheexcessof=signsover=signs(ortheotherwayround).

EarlyinFebruary2014AlexeiLisitsaandBorisKonevannouncedthattheanswertoErdoős’squestionis‘yes’ifC=2.Indeed,ifweselectad-stepsub-sequencefromthefirst1,161termsofany 1sequence,andchoosetheappropriatelengthk,theabsolutevalueofthesumexceedsC=2.Theirproofrequiresheavyuseofacomputer,andthedetailsrequirea13-gigabytedatafile.ThisismorethantheentirecontentsofWikipedia,at10gigabytes.Itiscertainlyoneofthelongestproofsever,andalreadytoolongforahumanbeingtocheck.

LisitsaisnowlookingforaproofforC=3,butthecomputerhasnotyetcompleteditscalculations.ItissoberingtothinkthatacompletesolutionrequiresunderstandingwhathappensforanychoiceofC.ThehopeisthatcomputersolutionsforsmallCmightrevealanewidea,whichahumanmathematiciancouldturnintoageneralproof.Ontheotherhand,theanswertoErdoős’squestionmightbe‘no’.Ifso,there’sareallyinterestingsequenceof1’soutthere,waitingtobedefined.

TheGreekIntegratorFromtheMemoirsofDrWatsup

Althoughmyfriend’sinvestigatorypowersaremainlydirectedtowardsthepursuitofcrime,fromtimetotimehisskillsareappliedintheserviceofscholarship.OnesuchinstancewasasingularquestthatwecarriedoutintheAutumnof1881atthebehestofawealthybutreclusivecollectorofancientmanuscripts.Withtheaidofatornpagefromanoldnotebook,alantern,abunchofskeletonkeys,andalargecrowbar,SoamesandIlocatedanenormousflagstoneandleveredituptouncoveraspiralstairwaythatleddowntoaconcealedchamberdeepbeneaththelibraryofafamousEuropeanUniversity.

Soamesconsultedthetornscrapofpaper,muchdamagedbyfireandwater.“TheLostIncunabulaoftheCartonari,”heexplained.

“Themagain!”HehadmentionedthisnameinpassingduringtheAdventureoftheCardboardBoxes(Seepage23),butdeclinedtosaymore.NowIpressedhimforfutherdetails.

“Thenamemeans‘cardboardmanufacturers’.ItisanItaliansecretsocietyorganisedalongthelinesoftheFreemasonsanddedicatedtothecauseofNationalism,beingimplicatedinthefailedrevolutionof1820.”

“Irecalltherevolutionwiththeutmostclarity,Soames.Butnottheorganisation.”

“Fewindeedareawareofitshiddenhand.”Heconsultedthescrapofpaper.“Thepageisalmostobliterated,butittakesnogreatexpertiseinhighermathematicstorecognisethatitissomeformofFibonaccicode,rewritteninDaVincimirrorscript,andtransformedintoasequenceofrationalpointsonanellipticcurve.”

“Evenachildcouldseethat,”saidI,lyingthroughmyteeth.“Quite.Now,ifIreadtheserunesaright,weshouldfindwhatweseek

somewhereontheseshelves.”Afteramoment,Iasked:“Soames,whatdoweseek?Youhaveplayedyour

cardsunusuallyclosetoyourchest.”“Itisknowledgethatholdsgreatdangers,Watsup.Isawnoneedtoexpose

youtoviolenceprematurely.Butnowthatwehavepenetratedtheinnersanctum—Ah!Hereitis!”HeextractedwhatIimmediatelyrecognisedtobeaparchmentcodex,blowingoffcenturiesofaccumulateddust.

“Whatthedevilisthat,Soames?”“Doyouhaveyourservicerevolver?”“Neverwithoutit.”“ThenitissafetotellyouthatinmyhandsIhold…theArchimedes

Palimpsest!”“Ah.”Now,Iwasawarethatapalimpsestisadocumentthathasbeenwrittenon

andthenscrubbedcleantoallowfurtherinscription,andthatscholarscanwithdifficultyreconstructthatwhichhasbeenobliterated,therebyretrievingahithertounknowngospelfromthelaundrylistofanobscureorderoffourteenth-centurymonks.Archimedeswasalsofamiliartome,asanancientGreekgeometerofprodigiousability.ItwasthereforeapparentthatSoameshadunearthedsomepreviouslyunknownmathematicaltext.Butheinsistedthatweshouldmakeourexitimmediately,beforetheInquisitorialVengeanceSquaddescendeduponus.

BackinthecomparativesafetyofBakerStreet,weinspectedthedocument.“Itisatenth-centuryByzantinecopyofahithertounknownworkof

Archimedes,”saidSoames.“ItstitlelooselytranslatesasTheMethod:itconcernsthatgeometer’scelebratedworkonthevolumeandsurfaceareaofasphere.Itshowsushowhecametodiscovertheseresults,offeringanunparalleledinsightintohisthoughtprocesses.”

Iwasstruckdumb,andnodoubtresembledagoldfishoutofwater.“Archimedesdiscoveredthatifasphereisinscribedinatightlyfitting

cylinder,thenthevolumeofthesphereisexactlytwothirdsthevolumeofthecylinder,anditsareaisthesameasthatofthecurvedsurfaceofthecylinder.Inmodernlanguage,iftheradiusisrthenthevolumeis andthearea4πr2.

“Now,Archimedeswassuchagreatmathematicianthathewasabletofindalogicallyrigorousgeometricproofofthesefacts,whichheincludedinhisbookOntheSphereandCylinder.Thereheusedacomplicatedmethodofproofnowknownasexhaustion.Oneofitstrickyfeaturesisthatonemustknowtheexactanswertotheproblembeforeprovingittobecorrect.Soithaslongbeenapuzzletoscholars:howdidArchimedesknowwhattherightanswermustbe?”

“Isee,”saidI.“Thislong-lostdocumentexplainshowhedidit.”“Exactly.Remarkably,itcomesclosetobeingananticipation,forthis

particularexample,oftheintegralcalculusofIsaacNewtonandGottfriedLeibniz,developedmorethantwothousandyearslater.But,asArchimedeswellknew,theideasusedinTheMethodlackrigour.Hencehisuseofexhaustion,averydifferentapproach.”

“Sohowdidhedoit?”Iasked.

Soamesstudiedthepalimpsestthroughhismagnifyingglass.“TheGreekisnotentirelyclassical,andoftenunclear,butthatposesnoseriousdifficultytoanexpertlinguistsuchasmyself.HaveIshownyoumypamphletonthedeciphermentofobscureancienttextsfromtheMediterraneanregion?Remindmetodoso.

“ItseemsthatArchimedesbeganwithasphere,acone,andacylinderofsuitabledimensions.Thenheimaginedtakingaverythinsliceofeach,andhangingthemonabalance:asliceofthesphereandasliceoftheconeononeside,asliceofthecylinderontheother.Ifthedistancesarechosencorrectly,themasseswillbalanceexactly.Sincemassisproportionaltovolume,thevolumesarerelatedbythelawofthelever.”

Slicingthesolidspriortohangingthemfromabalance:Seepage298fordetails

“Uh—prayremindmeofthislaw,”saidI.“Itwasunaccountablynotpartofthesyllabusatmedicalschool.”

“Itshouldhavebeen,”saidSoames.“Itwouldbeofgreatusewhendealingwithdislocatedjoints.Nomatter.Thelaw,whichwasdiscoveredandprovedbyArchimedes,statesthattheturningeffect,ormoment,ofagivenmassatagivendistanceisthemassmultipliedbythedistance.Forasetofmassestobalance,thetotalclockwisemomentmustequalthetotalanticlockwisemoment.Or,withappropriateassignmentofplusandminussigns,thetotalmomentmustbezero.”

“Er—”“Amassatagivendistancewillbalancehalfthatmassattwicethedistance,

provideditisontheotherarmofthebalance.”“Isee.”“Isuspectnot,butletmeproceed.Bysplittingthesesolidsintoinfinitely

manyinfinitelythinslices,andhangingthemappropriatelyonhisbalance,Archimedeswasabletoconcentratetheentiremassofthesphere,andthatofthecone,atasinglepoint.Theslicesofcylinder,whichareidenticalcircles,areplacedatdifferentdistances;togethertheyreconstructtheoriginalcylinder.Knowingthatthevolumeofthecone,henceitsmass,isonethirdthatofthecylinder,Archimedescouldthensolvetheresulting‘equation’forthevolumeofthesphere.”

“Amazing,”saidI.“Itseemsconvincingenoughtome.”

“ButnottoamathematicianoftheintellectualcalibreofArchimedes,”saidSoames.“Ifthesliceshavefinitethickness,theprocedureinvolvessmallbutunavoidableerrors.Butifthesliceshavezerothickness,theyhavezeromass.Thereisnouniquebalancepointwhenthemassesconcernedareallzero.”

Ibegantoseetheobjectiontotheprocedure.“Presumablytheerrorsbecomeeversmallerastheslicesbecomethinner?”Ihazarded.

“Theydo,Watsup,theydo.Andthemodernapproachtointegralcalculusconvertsthatstatementintoaproofthatthiskindofprocessgivessensibleanswers.However,theseideaswerenotavailabletoArchimedes.Soheusedanon-rigorousmethodtofindthecorrectanswer,andthatallowedhimtouseexhaustiontoprovetheanswerwascorrect.”

“Astonishing,”Isaid.“Wemustpublishthepalimpsest.”Soamesshookhishead.“AndriskthewrathoftheCartonari?Ivalueboth

ourlivestoohighlytoattracttheirattention.”“Sowhatshouldwedo?”“Wemustplacethemanuscriptsomewheresafe.Notbackinthelibrary,for

theymustbynowhavenoticeditsdisappearanceandsetsubtletraps.Iwillconcealitinsomeotherscholarlylibrary.No,donotaskwhich!Perhapsatsomelaterdate,whentimesarelesstroubledandtheinfluenceofsecretsocietieshaswaned,itwillberediscovered.Untilthen,wemustbecontentwithknowingthegreatgeometer’smethod,eventhoughwecannotrevealittotheworld.”

Hepaused.“Ihavealreadytoldyoutheformulasfortheareaandvolumeofasphere.Sohereisasimplelittleconundrumthatmightamuseyou.Whatshouldtheradiusofthespherebe,infeet,sothatitsareainsquarefeetisexactlyequaltoitsvolumeincubicfeet?”

“Ihavenoidea,”saidI.“Thenworkitout,man!”hecried.

Seepage296forthetruehistoryoftheArchimedespalimpsestandtheanswertoSoames’spuzzle.

SumsofFourCubesSumsoffoursquares,likemanymathematicalmysteries,havealonghistory.TheGreekmathematicianDiophantus,whoseArithmeticaofaboutAD250wasthefirsttextbooktouseaformofalgebraicsymbolism,askedwhethereverypositivewholenumberisasumoffourperfectsquares(0allowed).It’seasytoverifythisstatementexperimentallyforsmallnumbers;forexample,

5=22+12+02+026=22+12+12+027=22+12+12+12

Justwhenyouthinkthat8willneedanother12,hencefivesquares,4comestotherescue:

8=22+22+02+02

Experimentswithlargernumbersstronglysuggestthattheansweris‘yes’,buttheproblemremainedunsolvedforover1,500years.ItbecameknownasBachet’sproblem,afterClaudeBachetdeMéziriacpublishedaFrenchtranslationofArithmeticain1621.Joseph-LouisLagrangefoundaproofin1770.Simplerproofshavebeenfoundmorerecently,basedonabstractalgebra.

Whataboutsumsoffourcubes?Alsoin1770,EdwardWaringstatedwithoutproofthateverypositivewhole

numberisthesumofatmost9cubesand19fourthpowers,andaskedwhethersimilarstatementsaretrueforhigherpowers.Thatis,givenanumberk,istheresomefinitelimittothenumberofkthpowersneededtoexpressanypositivewholenumberbyaddingthemup?In1909DavidHilbertprovedthattheansweris‘yes’.(Oddpowersofnegativenumbersarenegative,andthatchangesthegameconsiderably,soforthemomentwerestrictattentiontopowersofpositivenumbers.)

Thenumber23definitelyrequires9cubes.Theonlypossibilitiesare8,1,and0,andthebestwecandois8+8+seven1’s:

23=23+23+13+13+13+13+13+13+13

Sowecan’tsucceedingeneralwithfewerthan9cubes.However,thatnumbercanbereducedifweignoreafinitenumberofexceptions.Forexampleonly23and239actuallyneed9cubes;allotherscanbeachievedusing8.YuriLinnikreducedthisto7byallowingafewmoreexceptions,anditiswidelybelievedthatthecorrectnumber,allowingfinitelymanyexceptions,is4.Thelargestknownnumberthatneedsmorethan4cubesis7,373,170,279,850,anditisconjecturedthatnolargernumberswiththatpropertyexist.Soitisverylikely—butremainsanopenquestion—thateverysufficientlylargepositivewholenumberisasumoffourpositivecubes.

But,asIsaidearlier,thecubeofanegativenumberisnegative.Thisallowsnewpossibilitiesthatdonotoccurforevenpowers.Forexample,

23=27–1–1–1–1=33+(–1)3+(–1)3+(–1)3+(–1)3

usingonly5cubes,whereaswithpositiveorzerocubesitrequires9,aswe’vejustseen.Butwecandobetter:23canbeexpressedusingjustfourcubes:

23=512+512–1–1000=83+83+(–1)3+(–10)3

Allowingnegativenumbersmeansthatthecubesinvolvedmightbemuchlarger(ignoringtheminussign)thanthenumberconcerned.Asanexample,wecanwrite30asasumofthreecubes,butwehavetoworkprettyhard:

30=2,220,422,9323+(–283,059,965)3+(–2,218,888,517)3

Sowecan’tworksystematicallythroughalimitednumberofpossibilities,aswecanwhenonlypositivenumbersareconsidered.

Experimentshaveledseveralmathematicianstoconjecturethateveryintegeristhesumoffour(positiveornegative)integercubes.Asyet,thisstatementremainsmysterious,buttheevidenceissubstantial.Computercalculationsverifythateverypositiveintegerupto10millionisasumoffourcubes.V.Demjanenkohasprovedthatanynumbernotoftheform9k 4isalwaysasumoffourcubes.

WhytheLeopardGotItsSpots

Leopardess,KananaCamp,Botswana

Leopardshavespots,tigershavestripes,andlionsareplain.Why?Itallseemsratherarbitrary,asiftheBigCatSalesCataloguelistscoatpatternsandevolutionpickswhicheverlooksprettiest.Butevidenceisaccumulatingthatit’snotlikethat.WilliamAllenandcolleagueshaveinvestigatedhowthemathematicalrulesthatdeterminethepatternsrelatetothecats’habitsandhabitats,andhowthisaffectsthewaythepatternsevolve.

Themostobviousreasonforevolvingpatternedcoatsiscamouflage.Ifacatlivesintheforest,spotsorstripeswillmakeithardtoseeamongthelightandshade.Catsthatoperateoutintheopen,ontheotherhand,willbeeasiertoseeiftheyhavestrongpatterns.However,theoriesofthiskindarelittlebetterthanjust-sostories,unlesstheycanbesupportedbyevidence.Experimentalverificationisdifficult:imaginepaintingoutatiger’sstripesforseveralgenerations,orfittingitanditsdescendantswithplainovercoats,toseewhathappens.Alternativetheoriesabound:markingsmightexisttoattractmates,ormerelybeanaturalconsequenceoftheanimal’ssize.

Themathematicalmodelofcatpatterningmakesitpossibletotestthecamouflagetheory.Somepatterns,suchastheleopard’sspots,areverycomplex,andthetypeofcomplexityiscloselyrelatedtothepattern’svalueascamouflage.Sotheresearchersclassifiedpatternsusingamathematicalscheme

inventedbyAlanTuring,inwhichthepatternislaiddownbychemicalsthatreacttogetheranddiffuseacrossthesurfaceofthedevelopingembryo.

Theseprocessescanbecharacterisedbyspecificnumbersthatdeterminetherateofdiffusionandthetypeofreaction.Thesenumbersactlikecoordinateson‘camouflagespace’,thesetofallpossiblepatterns,justaslatitudeandlongitudeprovidecoordinatesonthesurfaceoftheEarth.

Theresearchrelatesthesenumberstoobservationaldataon35differentcatspecies:whatkindofhabitatthecatsprefer,whattheyeat,whethertheyhuntbydayorbynight.Statisticalmethodsidentifysignificantrelationshipsbetweenthesevariablesandtheanimals’coatpatterns.Theresultsshowthatpatternsarecloselyassociatedwithclosedenvironments,suchasforests.Animalsinopenenvironments,suchassavannahs,aremorelikelytobeplain,likelions.Ifnot,theyusuallyhavesimplepatterns.Butcatsthatspendalotoftimeintrees,suchasleopards,aremorelikelytohavepatternedcoats.Moreover,thesepatternstendtobecomplex,notjustsimplespotsorstripes.Themethodalsoexplainswhyblackleopards(‘panthers’)arecommon,buttherearenoblackcheetahs.

Thedataargueagainstseveralalternativestocamouflage.Thesizeofthecatandthesizeofitspreyhavelittleeffectonpatterns.Sociablecatsarenomoreorlesslikelytobepatternedthansolitaryones,sothemarkingsareprobablynotimportantforsocialsignalling.Thisworkdoesnotprovethatcats’markingsevolvedforcamouflagealone,butitsuggeststhatcamouflageplayedakeyevolutionaryrole.

Lionsareplainbecausetheyprowlontheplains.Leopardsarespottybecausespotsarehardertospot.

Seepage299forfurtherinformation.

PolygonsForever

Keepgoingforever…Howbigdoesitget?

Here’satestofyourgeometricandanalyticintuition.Startwithacircleofunitradius.Drawthetightestfittingequilateraltrianglethatyoucanroundit;thendrawthetightestfittingcircleroundthat.Repeat,butatsuccessivestagesuseasquare,aregularpentagon,aregularhexagon,andsoon.

Ifthisprocessgoesonforever,doesthepicturebecomearbitrarilylarge,ordoesitremainwithinaboundedregionoftheplane?

Seepage299fortheanswer.

TopSecretInthe1930saRussianmathematicsprofessorwasrunningaseminarinfluiddynamics.Twooftheregularattendees,alwaysdressedinuniform,wereobviouslymilitaryengineers.Theyneverdiscussedtheprojecttheywereworkingon,whichwaspresumablytopsecret.Butonedaytheyaskedtheprofessorforhelpwithamathematicalproblem.Thesolutionofacertainequationoscillated,andtheywantedtoknowhowtochangethecoefficientstomakeitmonotonic.

Theprofessorlookedattheequationandsaid:“Makethewingslonger!”

TheAdventureoftheRowingMenFromtheMemoirsofDrWatsup

IhaveoftenbeenastonishedbySoames’sabilitytoperceivepatternsinthemostunpromisingofcircumstances.Nobetterexamplecouldbefoundthanthatwhichoccurredintheearlyspringof1877.

AsIwalkedacrossEquilateralParktowardshislodgings,afreshlymintedsuncastdappledlightandshadethroughascatteringoffluffyclouds,andthehedgerowsrangwithbirdsong.Onsogloriousadayitseemedveritablyindecenttoremainindoors,butmyeffortstodragmyfriendawayfromcataloguinghiscomprehensivecollectionofusedmatchsticksmetwithindifference.

“Manyacasehashingeduponthetimeittookamatchtoburn,Watsup,”hegrumbled,transferringameasurementfromhisdividersintoanotebook.

Disappointed,Iopenedthenewspaperatthesportspages,andmyeyewascaughtbyatimelyreminderofaneventthatevenSoameswouldnotwishtomiss.Ithadcompletelyescapedmymindamidthebuzzingofthebeesandtheblossomingtrees.WithinthehourthetwoofuswereseatedontheriverbankwithaluncheonbasketandseveralbottlesofapalatableBurgundy,awaitingthestartoftheannualrace.

“Whomdoyoufavour,Soames?”HestoppedmeasuringthelengthoftheburnmarkonanearlyScottish

lucifer,forhehadinsistedonbringinganumberofmatchstickswithhimtohelppassthetime.“Theblueteam.”

“Dark,orlight?”“Yes,”hesaidenigmatically.“Imean:OxfordorCambridge?”“Yes.”Heshookhishead.“Oneofthose.Thevariablesaretoocomplexto

makeaprediction,Watsup.”“Soames,myquerywasaboutsupport,notprediction.”Hegavemeascathinglook.“Watsup,whyshouldIsupportmenwithwhom

Iamnotacquainted?”WhenSoamesisinamood,thereisalwaysareason.Inoticedthathewas

layingoutmatchsticksinpatternsresemblingthebonesonakipper,andaskedhimwhy.

“Ihavebeenobservingthedistributionoftheoars,andIamwonderingwhysuchaninefficientarrangementhasbecometraditional.”

IlookedatthetwoboatsastheylinedupontheThamesfortheannualUniversityBoatRace.“Traditionisofteninefficient,Soames,”Ichided.“Foritconsistsofdoingthingsthesamewaytheyhavealwaysbeendone,insteadofaskinghowbesttodothem.ButIseenoinefficiencyhere.Thereareeightrowers,andtheoarspointalternatelytotheleftandtheright.Itisknownastandemrigging.Thatseemssymmetricandsensibletome.”

Tandemrigging(arrowshowsbowofboat)

Soamesgaveadissatisfiedgrunt.“Symmetric?Pah!Notatall.Theoarsononesideoftheboatareallinfrontofthecorrespondingoarsontheotherside.Sensible?Theasymmetrycreatesatwistingforcewhentherowerspullontheoars,causingtheboattoveertooneside.”

“That,Soames,isonereasonwhythereisacoxswain.Whohasarudder.”“Whichcreatesresistancetotheforwardmotionoftheboat.”“Oh.Buthowelsecouldtheoarsbearranged?Itisnotpossibletosittwo

rowerssidebyside.”“Thereare68alternatives,Watsup;34ifwecountleft–rightreflectionsas

beingthesame.OurGermanandItalianfriendsusedifferentarrangements,tobespecific.”Helaidouttwoskeletalarrangementsofmatchsticks.

Left:Germanrigging.Right:Italianrigging.

Istaredatthem.“Surelysuchstrangearrangementssufferfromevenworseproblems!”

“Perhaps.Letussee.”Hepursedhislips,deepinthought.“Thereareinnumerablepracticalissues,Watsup,whichwouldrequireamorecomplexanalysis.Nottomentionmorematchsticks.SoIshallcontentmyselfwiththesimplestmodelIcandevise,inthehopeofgainingusefulinsights.Iwarnyounow,theresultswillnotbedefinitive.”

“Fairenough,”saidI.

Resolvingtheforces.NotethatPpointsforwardsandRpointsawayfromthecentreoftheboatbecausetheoutermostendoftheoarisheldfixed(toagoodapproximation)bytheresistanceofthewater.Don’tforget

thattherowerfacesthesternandpullstheoartowardshim.

“Letusthenconsiderasingleoar,andcalculatetheforcesactingontherowlockwhereitpivots,duringthatpartofthestrokewhentheoarisinthewater.ForsimplicityIshallassumethatallrowershavethesamestrength,androwinperfectsynchrony,sotheyexertidenticalforcesFatanygiveninstant.IthenresolvethisforceintoacomponentPparalleltotheaxisoftheboat,andRatrightangles.”

“Alloftheseforcesvarywithtime,”saidI.Henodded.“Whatmattersiswhatmechanicianscallthemomentofeach

force—theextenttowhichitturnstheboataboutsomechosenpoint.This,youwillrecallfromourencounterwiththeArchimedesPalimpsest,isfoundbymultiplyingtheforcebyitsperpendiculardistancefromthatpoint.”

Itwasmyturntonod.IwassureIhadrememberedsomethingofthekind.“Imarkthepositionofthesternmostoarbyadot.Thiswillbeourchosen

point.Now,theforcePhasmomentPdaboutthepointatwhichtherowlockmeetsthecentralaxisoftheboat,iftheoarisontheleft-handside.Butifitisontheright-handside,themomentis–Pdsincetheforceactsintheoppositedirection.Noticethatthesemomentsarethesameforallfouroarsonthesamesideoftheboat.Inconsequence,thetotalmomentofalleightoarsis4Pd–4Pd,whichis0.”

“Thetwistingforcescancelout!”“FortheparallelforcesP,yes.However,themomentoftheforceRis

differentforeachoar,foritdependsonthedistancexbetweenthatoarandtheoneatthestern.Infact,itisRx.Ifsuccessiveoarsareseparatedbythesamedistancec,thenxtakesthevalues

0cR2cR3cR4cR5cR6cR7cR

astheoarsrunfromthesterntothebow.Thereforethetotalmomentis

0 cR 2cR 3cR 4cR 5cR 6Rc 7cR

wherethesignisplusforoarsontheleftoftheboat,butminusforthoseontheright.”

“Why?”“Forcesontheleftsideturntheboatclockwise,Watsup,whereasthoseon

therightturnitanticlockwise.Wecansimplifythisexpressionto

( 0 1 2 3 4 5 6 7)cR

wherethepatternofplusandminussignsmatchesthesequenceofsidesatwhichtheoarsareplaced.

“Nowconsidertandemrigging.Herethesequenceofsignsis

+–+–+–+–

sothecombinedturningmomentis

(0–1+2–3+4–5+6–7)cR=–4cR

Duringthefirstpartofthestroke,Rpointsinwards,butoncetheoarstartstotrailbackwardsthedirectionofRreversesanditpointsoutwards.Sotheboatfirstturnsinonedirection,thenintheother,creatingawigglingmotion.Thecoxhastousetheruddertocorrectthis,and—asIhavesaid—thiscreatesresistance.

“WhatoftheGermanrigging?Nowthecombinedturningmomentis

(0–1+2–3–4+5–6+7)cR=0

whatevercandRmightbe.Sothereisnotendencytowiggle.”“WhatoftheItalian?”Icried.“Oh,doletmetry!Thecombinedturning

momentis

(0–1–2+3+4–5–6+7)cR=0

aswell!Howremarkable.”“Quite,”repliedSoames.“Now,Watsup,hereisaquestionforyouragile

mind.AretheGermanandItalianrigs—ortheirleft–rightreflections,whichdifferonlytrivially—theonlywaystomaketheturningforcesequalzero?”Hemusthaveseenthelookonmyface,forheadded:“Thequestionboilsdowntoseparatingthenumbersfrom0to7intotwosetsoffour,eachhavingthesame

sum.Whichmustbe14sinceallsevennumbersaddupto28.”

Seepage300fortheanswer,andfortheresultofthe1877BoatRace.

TheFifteenPuzzleThisisanoldfavourite,butnonetheworseforthat.It’safascinatingcasewherealittlemathematicalinsightcouldhavesavedanawfullotofwastedeffort.Plus,Ineedittosetupthenextitem.

In1880aNewYorkpostmasternamedNoyesPalmerChapmancameupwithwhathecalledtheGemPuzzle,andthedentistCharlesPeveyofferedmoneyforasolution.Thistriggeredabriefcraze,butnoonewonthecash,soitquicklydieddown.TheAmericanpuzzlistSamLoyd*claimedthathehadstartedacrazeforthispuzzleinthe1870s,butallhereallydidwastowriteaboutitin1896,offeringaprizeof$1,000,whichrevivedinterestforatime.

Thepuzzle(alsocalledtheBossPuzzle,GameofFifteen,MysticSquare,andFifteenPuzzle)startswith15slidingblocksnumbered1–15arrangedinasquare,withanemptysquareatbottomright.Blocksareinnumericalorder,exceptfor14and15.Yourtaskistoswapthe14and15,leavingeverythingelseasitwas.Youdothisbymovinganyadjacentblockintotheemptysquare,andrepeatingthesemovesasoftenasyouwish.

Asyoumovemoreandmoreblocks,thenumbersgetjumbledup.Butifyou’recareful,youcanunjumblethemagain.It’seasytoassumethatanyarrangementcanbeobtainedifyou’recleverenough.

Fifteenpuzzle.Left:Start.Middle:Finish.Right:Colouringtheblocksforanimpossibilityproof.

Footnote

*That’snotatypo:hedidn’tspellitwithadoubleL.

Loydwashappytooffersuchagenerous(atthetime)prize,becausehewasconfidenthewouldneverhavetopayup.Thereare16!potentiallypossible

arrangements:allpossiblepermutationsoftheblocks(15numberedplusoneempty).Thequestionis:whichofthesearrangementscanbereachedbyaseriesoflegalmoves?In1879WilliamJohnsonandWilliamStoryprovedthattheanswerisexactlyhalfofthem;and—wouldn’tyoujustknowit—thearrangementthatgetsthecashisintheotherhalf.TheFifteenPuzzleisinsoluble.Butmostpeopledidn’tknowthat.

Theimpossibilityproofinvolvescolouringthesquareslikeachessboard,asintheright-handfigure.Slidingablockineffectswapsthatblockwiththeemptysquare,andeachswapchangesthecolourassociatedwiththeemptysquare.Sincetheemptysquaremustendupinitsoriginalposition,thenumberofswapsmustbeeven.Everypermutationcanbeobtainedfromsomeseriesofswaps;however,halfofthemuseanevennumberofswapsandtheotherhalfuseanoddnumber.

Therearemanywaystoachieveanygivenpermutation,buttheyareeitheralloddoralleven.Thedesiredresultcouldbeobtainedwithjustoneswap,interchanging14and15,butoneisodd,soyoucan’tachievethispermutationwithanevennumberofswaps.

Thisconditionturnsouttobetheonlyobstacle:legalmovesleadtoexactlyhalfofthe16!possiblerearrangements.Now16!/2=10,461,394,944,000;suchalargenumberthathowevermanytimesyoutry,mostpossibilitiesremainunexplored.Thiscouldencourageyoutothinkthatanyarrangementmustsurelybepossible.

TheTrickySixPuzzleIn1974RichardWilsongeneralisedtheFifteenPuzzleandprovedaremarkabletheorem.Hereplacedtheslidingblocksbyanetwork.Theblocksarerepresentedbynumbersthatcanbeslidalonganedge,provideditisconnectedtothenodecurrentlybearingtheblanksquare.Theblanksquarethenmovestoanewlocation.Thefigureshowsthestartingpositionoftheblocks.Nodesarelinkedifthecorrespondingsquaresareadjacent.

NetworkrepresentingtheFifteenpuzzle

Wilson’sideaistoreplacethisnetworkbyanyconnectednetwork.Supposeithasn+1nodes.Initially,onenode,markedbythebox,isempty(fromnowon,thinkofitasnode0)andtheresthaveanumber(1–n)sittingonthem.Thepuzzleamountstomovingthesenumbersaroundthenetwork,byswapping0withthenumberonanadjacentnode.Therulesspecifythat0mustendupwhereitstarted.Theothernnumberscanbepermutedinn!ways.Wilsonasked:whatfractionofthesepermutationscanbeachievedbylegalmoves?Theanswerclearlydependsonthenetwork,butnotasmuchasyoumightexpect.

Thereisoneobviousclassofnetworksforwhichtheanswerisunusuallysmall.Ifthenodesformaclosedring,theinitialarrangementistheonlyoneyoucanreachbylegalmoves,because0hastoreturntoitsstartingpoint.Alltheothernumbersstayinthesamecyclicorder;there’snowayforanumbertoworkitswayroundthesideofanotherone.RickWilson’sTheorem(sonamedtoavoidconfusionwithanothermathematicalWilson)statesthatasidefromclosedrings,eitherallpermutationscanbeachieved,orexactlyhalfofthem(theevenones).

Withexactlyonegloriousexception.

Thetheoremrevealsasurprise.Auniquesurprise:anetworkwithsevennodes.Sixformahexagon,andtheotheronesitsinthemiddleofadiameter.Thereare6!=720permutations;halfofthisis360.Buttheactualnumberthatcanbereachedisonly120.

Wilson’sexceptionalnetwork

Thereasoninginvolvesabstractalgebra,namelysomeelegantpropertiesofgroupsofpermutations.Detailsarein:AlexFinkandRichardGuy,Rick’strickysixpuzzle:S5sitsspeciallyinS6,MathematicsMagazine82(2009)83-102.

AsDifficultasABC

Fromtimetotimemathematicianshaveapparentlycrazyideasthatturnouttohavehugeimplications.TheABCConjectureisoneofthem.RememberFermat’sLastTheorem?In1637PierredeFermatconjecturedthat

ifn≥3therearenononzerointegersolutionstotheFermatequation

an+bn=cn

Ontheotherhand,thereareinfinitelymanysolutionswhenn=2,suchasthePythagoreantriple32+42=52.Ittook358yearstoproveFermatwasright,withtheworkofAndrewWilesandRichardTaylor(seeCabinetpage50).Jobdone,youmightthink.Butin1983RichardMasonrealisedthatnoone

hadlookedcloselyatFermat’sLastTheoremforfirstpowers:

a+b=c

Youdon’tneedtobeawhizzatalgebratofindsolutions:1+2=3,2+2=4.ButMasonwonderedwhetherthequestiongetsmoreinterestingifyouimposedeeperconditionsona,b,andc.Whateventuallyemergedwasashiningnewconjecture,theABC(orOesterlé–Masser)Conjecture,whichwillrevolutionisenumbertheoryifanyonecanproveit.It’ssupportedbyavastquantityofnumericalevidence,butaproofseemselusive,withthepossibleexceptionofworkofShinichiMochizuki.I’llgetbacktothatonceweknowwhatwe’retalkingabout.Morethantwothousandyearsago,EuclidknewhowtofindallPythagorean

triples,usingwhatwewouldnowwriteasanalgebraicformula.In1851JosephLiouvilleprovedthatnosuchformulaexistsfortheFermatequationwhenn≥3.Masonwonderedaboutthesimplerequation

a(x)+b(x)=c(x)

wherea(x),b(x),andc(x)arepolynomials.Apolynomialisanalgebraiccombinationofpowersofx,like5x4–17x3+33x–4.Again,it’seasytofindsolutions,buttheycan’tallbe‘interesting’.The

degreeofapolynomialisthehighestpowerofxthatoccurs.Masonprovedthatifthisequationholds,thedegreesofa,b,andcarealllessthanthenumberofdistinctcomplexsolutionsxoftheequationa(x)b(x)c(x)=0.Itturnedoutthat

W.WilsonStothershadprovedthesamethingin1981,butMasondevelopedtheideafurther.Numbertheoristsoftenlookforanalogiesbetweenpolynomialsandintegers.

ThenaturalanalogueoftheMason–Stotherstheoremwouldbethis.Supposethata+b=c,wherea,b,andcareintegerswithnocommonfactor.Thenthenumberofprimefactorsofeachofa,b,andcislessthanthenumberofdistinctprimefactorsofabc.Unfortunately,thisisspectacularlyfalse.Forexample,14+15=29,which

hasonlyone(necessarilydistinct)primefactor,whereas14=2×7and15=3×5bothhavetwoprimefactors.Oops.Undaunted,mathematicianstriedtomodifythestatementuntilitlookedlikeitmightbetrue.In1985DavidMasserandJosephOesterlédidjustthat.Theirversionstates:

Foreveryε>0,thereareonlyfinitelymanytriplesofpositiveintegers,withnocommonfactors,satisfyinga+b=c,suchthatc>d1+ε,whereddenotestheproductofthedistinctprimefactorsofabc.

ThisistheABCConjecture.Ifitweretobeproved,manydeepanddifficulttheorems,provedoverpastdecades,withenormousinsightandeffort,wouldbedirectconsequences,andthereforehavesimplerproofs.Moreover,allofthoseproofswouldbeverysimilar:dosomeminorandroutinesettingup,andthenapplytheABCTheorem,asitwouldthenbecome.AndrewGranvilleandThomasTucker[It’saseasyasabc,NoticesoftheAmericanMathematicalSociety49(2002)1224–1231]writethataresolutionoftheconjecturewouldhave“anextraordinaryimpactonourunderstandingofnumbertheory.Provingordisprovingitwouldbeamazing.”BacktoMochizuki,awell-respectednumbertheoristwithasolidtrackrecord

ofresearch.In2012heannouncedaproofoftheABCConjectureinaseriesoffourpreprints—papersnotyetsubmittedforofficialpublication.Contrarytohisintentions,thisattractedtheattentionofthemedia,thoughitwassurelyunrealistictoimaginethatthiscouldpossiblyhavebeenavoided.Expertsarecurrentlycheckingthe500orsopagesofradicallynewmathematicsinvolved.Thisistakingalotoftimeandeffortbecausetheideasaretechnical,complicated,andunorthodox;however,nooneisrejectingitbecauseofthat.Oneerrorhasbeenfound,butMochizukihasstatedthatitdoesnotaffecttheproof.Hecontinuestopostprogressreports,andtheexpertscontinuechecking.

RingsofRegularSolids

Eightidenticalcubesfittogether,facetoface,tomakeacubetwicethesize.Eightcubesalsofittogethertoforma‘ring’—asolidwithaholeinit,topologicallyatorus.

Aringofcubes

Withabitmoreeffort,youcandothesamewiththreeotherregularsolids:theoctahedron,dodecahedron,andicosahedron.Inallfourcasesthesolidsareexactlyregularandtheyfittogetherexactly:thisisobviousforcubesandisasimpleconsequenceofsymmetryfortheotherthreesolids.

Ringsofoctahedrons,dodecahedrons,andicosahedrons

However,therearefiveregularsolids,andthismethoddoesn’tworkfortheoneremainingtype,atetrahedron.Soin1957HugoSteinhausaskedwhetheranumberofidenticalregulartetrahedronscanbegluedface-to-facesothattheyformaclosedring.Hisquestionwasansweredayearlater,whenS.Świerczkowskiprovedthatsuchanarrangementisimpossible.Tetrahedronsarespecial.However,in2013MichaelElgersmaandStanWagondiscoveredabeautiful

eightfoldsymmetricringmadefrom48tetrahedrons.WasŚwierczkowskiwrong?

ElgersmaandWagon’sring.Left:Perspectiveview.Right:Viewfromabovetoshoweightfoldsymmetry.

Notatall,asElgersmaandWagonexplainedintheirarticleabouttheirdiscovery.Ifyoumaketheirarrangementusinggenuinelyregulartetrahedrons,theyleaveasmallgap.Youcanclosethegapbyelongatingtheedgesshownasthickerlines,from1unitto1.00274,adifferenceofonepartinfivehundred,whichthehumaneyecan’tdetect.

Thegap,exaggerated

Świerczkowskiasked:ifyouuseenoughtetrahedrons,andfitthemtogetherinaringtoleaveagap,howsmallcanthegapbe?Canyoumakeitassmallasyouplease,relativetothesizeofasingletetrahedron,byusingsufficientlymany?Theanswerisstillnotknownifthetetrahedronsarenotallowedtointersect,butElgersmaandWagonprovethattheansweris‘yes’iftheycaninterpenetrate.Forexample,438ofthemleaveagapthatisaboutonepartintenthousand.

ElgersmaandWagon’s438interpenetratingtetrahedrons

Theyconjecturethattheanswerremains‘yes’evenwhenthetetrahedronsarenotallowedtointersect,butthearrangementshavetobemorecomplicated.Asevidence,theyhavefoundaseriesofringswithever-narrowergaps.Thecurrentrecord,discoveredin2014,isanalmost-closedringof540non-intersectingtetrahedronswithgap5×10–18.Seepage302forfurtherinformation.

ElgersmaandWagon’sringof540non-intersectingtetrahedrons.

TheSquarePegProblem

Thismathematicalmysteryhasbeenopenformorethanacentury.Isittruethateverysimpleclosedcurveintheplane(onethatdoesnotcrossitself)containsfourpointsthatarethecornersofasquarewhosesideisnotzero?

Asimpleclosedcurveandasquarewhosecornerslieonit

‘Curve’hereimpliesthatthelineiscontinuous,nobreaks,butitneednotbesmooth.Itcanhavesharpcorners,andindeeditcanbeinfinitelywiggly.Weinsistthatthesideofthesquareisnonzerotoavoidthetrivialanswer:choosethesamepointforallfourcorners.ThefirstprintedreferencetotheSquarePegProblemappearedin1911ina

reportonaconferencetalkbyOttoToeplitzthatapparentlyclaimedaproof.However,noproofwaspublished.In1913ArnoldEmchprovedthestatementistrueforsmoothconvexcurves,sayingthathehadn’theardaboutitfromToeplitz,butfromAubreyKempner.Thestatementhasbeenprovedcorrectforconvexcurves,analyticcurves(definedbyconvergentpowerseries),sufficientlysmoothcurves,curveswithsymmetry,polygons,curveswithnocuspsandboundedcurvature,star-shapedtwice-differentiablecurvesthatmeeteverycircleinfourpoints…Yougetthepicture.Lotsoftechnicalhypotheses,nogeneralproof,no

counterexample.Perhapsit’strue,perhapsnot.Whoknows?Therearegeneralisations.TheRectangularPegProblemaskswhetherforany

realnumberr≥1,everysmoothsimpleclosedcurveintheplanecontainsthefourverticesofarectanglewithsidesintheratior:1.Onlythesquarepegcaser=1hasbeenproved.Thereareafewextensionstohigherdimensionsunderverystrongconditions.

TheImpossibleRoute

FromtheMemoirsofDrWatsup

Itiswithaheavyheart…Ithrewdownmypen,overcomewithgrief.ThatDevil’sspawn!Professor

Mogiarty’smachinationshadcausedtheprematuredemiseofoneofthegreatestdetectivesevertohavelimpedthestreetsofLondondisguisedasanelderlyRussianfishmonger.ThefinestmindIhaveeverencountered,snuffedoutbyacriminalwho—untilSoamesdispatchedhimatsuchcost!—hadafingerineveryfouldeedinthekingdom.Exceptfortheidiotwhooftenparkshiscarriagedirectlybeneathourwindow,wherehishorse—Pleasebearwithyourhumblescribeashewipesawayamanlyteartorecount

thetragicevents.Soameshadbeeninablackmoodforaweek.ItwaswhenIsawhimfitting

thesixthpadlocktothewindowandliningupthethirdGatlinggunthatIbegantosuspecthismindwastroubledinsomemanner.“Youmightsaythat,”saidhe.“Sowouldyoursbeifyouhadnarrowly

dodgedafallinggrandpianoonyourwaytothebarber’s—aChickering,bytheway,Icouldtellinstantlyfromthecast-ironframe.BeforeIhadgatheredmywits,Iwasforcedtoleapasidefromarunawaybrewer’sdraydrawnbyfourcarthorses,whichexplodedasplitsecondafterIhadhadtheforesighttotakecoverbehindaconvenientwall.Thepromptcollapseofthewallintoadeepcavitycameclosetodisturbingwhatlittleequilibriumremainedtome,butImanagedtoswingmyselftosafetyusingagrappling-ironthatIhabituallycarryinmypocketforsucheventualities.Itfoldsupforconvenience,andthecordislightbutstrong.Afterthat,thingsbecameatriflefraught.”IfIhadnotknownmyfriendbetter,Iwouldhavethoughthewasrattled.“Hasitoccurredtoyou,Soames,thatperhapssomeoneistryingtodoyou

harm?”Hesnortedinadmirationatmyastuteness,orsoIpresumed.“ItisMogiarty,”

hestatedflatly.“ButthistimeIhavethemeasureofhim.Evenaswespeakmycunningplaniscomingtofruition,andeverypolicemaninLondonisdescendingonthat…Wellingtonofcrime…andhisminions.Soontheywillbebehindbars,andthen…therope!”

Therewasaknockatthedoor.Anurchinappeared.“Telegramfer‘isnibs!”Soamestookthepaperandhandedtheurchinathruppenybit.“Thegoingrateissixpence,”saidtheurchin.“Whosaysso?”“’imacrosstheroad,guv.ThatMrSher—”“It’llbetuppenceandacliproundtheearifyoudon’tgoaway,”saidSoames.

Theurchinleft,mutteringunderhisbreath.Soamesopenedthefoldedpaper.“Nodoubtnewsoftheoperation’ssucce…”Hisvoicetrailedoff.“Whatisit?”Iaskedanxiously.Hisfacehadgonedeathlypale.“Mogiartyhasescaped!”“How?”“Disguisedasapoliceman.”“Thecunningfiend!”“ButIknowwherehehasgone,Watsup.Youhavetenminutestogohome

andpack.Thenwewillbetakingthecross-channelferry,assortedtrains,abrougham,adog-shay,anomnibus,andtwodonkeys.Oneeach.”“But—Soames!BeatrixandIhavebeenmarriedforlessthanamonth!I

cannotleave—”“Yournewbridewillhavetogetusedtothiskindofthingeventually,

Watsup,ifwearetocontinueourcollaboration.”“True,but—”“Notimelikethepresent.Absencemakestheheartgrowfonder.Adogisa

man’sbest—well,enoughclichés.Herbrotherwilltakecareofherwhileyouaregone.Anabsenceofsixweeksshouldbeample.”Irealisedthathewouldnotaskthisofmewithoutacompellingreason.He

neededme.Imustrisetotheoccasion,nomatterthepersonalcost.“Verywell,”Isaid,direforebodingsnotwithstanding.“Beatrixwillunderstand.Wherearewegoing?”“TotheSchtickelbachFalls,”hesaidquietly.Igaveaninvoluntaryshudder.Itwasanametostriketerrorintotheheartof

eventhemostaccomplishedmountaineer.“Soames!Thatissuicide!”Heshrugged.“ItiswhereweshallfindMogiarty.Butfirstwemustgetthere.”

Hepulledoutamap.

Soames’smap

“ThemapshowstheappropriateregionofSwitzerland.Observethenetworkofrivers.Theyriseinthenorthandflowacrossthecountry’sborders.TheSchtickelbachFallsareattheendofasmallriverbranchingoffalargerone.”“WheredoestherivergoaftertheFalls?”“Itplungesbeneaththeearth,intosomeundergroundpassage.Nooneknows

whereitreappears.”“Itisstrangegeology,Soames.”“TheSwisslandscapeisatorturedone,Watsup.Now,therearesixbridges,

whichIhavemarkedA,B,C,D,E,F.ThesearetheonlybridgeswithinSwissbordersthatjointheareasoflandshown.TheomnibusterminusisatthesmalltownofFroschma¨usekrieg.FromtherewewillhiredonkeysandproceedtotheFalls.WemuststayinSwitzerland:itwillbehardenoughtocrossanationalborderunnoticedonce,anditwouldbefoolhardyintheextremetorepeattheattempt.Ihavealreadyworkedoutaroute,butyoumayhaveabetteridea.”Istudiedthemap.“Why,itissimple!WecrossbridgeA.”“No,Watsup.Itistoodirect.Mogiartywillbeexpectingthat,itisabridgetoo

far.WemustleavebridgeAuntillastinthehopeofthrowinghimoffthescent.Andwemustcrosseachbridgeatmostoncetominimisethechanceofattractingattentionandbeingidentified.”“ThenwemustbeginwithbridgeB,”saidI.“TheonlycontinuationisviaC,

thenD.AtthatpointwehaveachoiceofEorF.BothleadtotheFalls,sowemayaswelluseE.Done!”“AsIsaid,wemustleaveAuntillast.NotE.”“Oh,yes.ThenweproceedacrossA—no,thatisadeadendwithnofurther

connectiontotheFalls.SoweleaveAforlaterandcrossF…Butno:thatalsoisadeadend.”

TwopathsthatdonotreachtheFalls

Soamesgruntednon-committally.Icheckedmyanalysis.“PerhapsbridgeF…no.ThesameproblemsariseifweuseFinsteadofE,aftercrossingD.Thereisnosuchpath,Soames!”Athoughtstruckme.“Unlessthereisatunnel,orsomeotherwaytocrossoneoftherivers.Aferry?Acanoe?”“Thereisnotunnel,orferry,orcanoe,andwedonotneedtocrossanyrivers.

Bridgesanddrylandsuffice.”“Thenthethingisimpossible,Soames!”Hesmiled.“ButWatsup:Ihavealreadytoldyouthatthereisaroute

satisfyingthestatedconditions.Indeed,therearenolessthaneightessentiallydifferentroutes—bywhichImeanthatthebridgesarecrossedinadifferentorder.”“Eight?IconfessIdonotseeevenone,”saidI,exasperated.

IsSoamesright?Seepage303fortheanswer.

TheFinalProblem

FromtheMemoirsofDrWatsup

Isleptbadlyandawokeatsunrise,tofindSoamesalreadydressed,bright-eyed,andbushy-tailed.“Timeforbreakfast,Watsup!”hedeclaredinaheartyvoice.Ifhewasapprehensiveabouttheforthcomingencounter,heconcealeditperfectly.Assoonaswehadfinishedourplatesofbread,meat,andSwisscheese,we

mountedourdonkeysandmadeourwayupanarrowtrack.Aftersomemileswetetheredourtrustysteeds,havingreachedthebaseoftheSchtickelbachFalls.Awildtorrentofwaterplungedbetweenthesheersidesofatoweringchasm,todisappearintoadeepholeintheground,creatingagloriousrainbowthatglitteredintheafternoonsun.AsteeprockypathledtothetopoftheFalls.Asweapproached,asilhouette

appearedontheskylineaboveus.“Mogiarty,”saidSoames.“Therecanbenomistakingthatevilprofile.”He

tookouthispistolandflickedoffthesafety-catch.“Thefiendistrapped,forthereisnowaydownsavethispathway.Notanypaththatamancantakeandstilllive,atanyrate.Waithere,Watsup.”“Nay,Soames!Iwillaccomp—”“Youwillnot.Thedutytoridtheworldofthisvilecreatureisminealone.I

willsignalwhenitissafeforyoutojoinme.Promisemeyouwillremainhereuntilyoureceiveit.”“Whatsignal?”“Youwillknowwhenthetimecomes.”Iassented,despiteprofoundmisgivings,andheascended,quickly

disappearingfromviewbehindarockycrag.ThelastIsawofhimwasapairofstoutclimbingboots.Iwaited.Allwassilence.Then,suddenly,Iheardshouting.ThewindcarriedawaythewordsandI

couldnotmakethemout.ThenIheardtheindubitablesoundsofaprolongedstruggle,andseveralgunshots.Therewasascream,andsomethingplungedpastmeamidthetorrent.ItwasshroudedbysprayandmovedsorapidlythatIcouldnotmakeitout,butitwasroughlythesizeofaman.Ortwomen.

Shockedtothecore,IneverthelessdidasSoameshadtoldme,andwaited.Nosignalcame.AtlastIdecidedthatsomethinghadgonewrong,whichrelievedmeofmy

promise.Iclamberedupthepath.Atthetop,therockclimbedfurtherskywardsinahugeoverhang,barringmyway.Amossyledgeledtowardstheprecipicefromwhichthewaterfallplunged.OfSoamesandMogiartytherewasnosign.But,dampenedbythespray,themossheldfainttracesoffootprints.Theprintstoldacleartaletoanyonewhohadabsorbedlessonsindetection

fromthemaster.IrecognisedtheindistinctimpressionswithachevronpatternasSoames’sboots;theotherprintswithazigzagwereevidentlyMogiarty’s.Thetwosetsofprintsledtothelipofthechasm,andherethegroundwaschurnedintothickmudbywhathadevidentlybeenthestruggleIhadheard.Isuckedinmybreathinhorror,fornofootprintsreturnedfromthatterrible

brink.IretainedenoughpresenceofmindtoattemptwhatSoameswouldhavedone,

facedwithsuchevidence.Beingcarefulnottosuperimposemyownfootprints—forthelocalconstabulary,ineptastheywouldundoubtedlybe,wouldnodoubtwishtoinspectthescene—Imadeathoroughstudy.SoameshadclearlywalkedbehindMogiarty,forhisprintssometimes

overlaidthoseofthecriminal,butnotviceversa.Mogiarty’sprintsseemeddeeperthanSoames’s,butthen,myfriendwasalwayslightoffoot.Thedismalconclusionwasclear.SoameshadpursuedMogiartytotheedgeoftheprecipice;therehadbeenastruggle;bothmen,nodoubtstillgrappling,hadplungedtotheirdoom.Theirbodieswerenowdeepundergroundinsomedankchamber,nevertoberecovered.Itrudgeddespondentlybacktothepath,wherethebarerockshowedno

footprints.Theclifftoweredaboveme,unscalable.IreasonedthathadSoamesprevailed,hewouldhavesignalledandbeawaitingmyarrival.HadMogiartyprevailed,hewouldhavebeenawaitingmeinstead,armedtotheteeth.Therewasnoconceivabledoubtthatbothmenhadmetthesameterribleend.Yet,evenasIbeganmydescent,myfriend’svoiceseemedtoechoinmy

mind,andthetonewasmocking.Wasmysubconscioustryingtotellmesomething?Griefovercamemyanalyticalabilities,andItrudgeddownhilltowardsthedonkeys.BywhichIrefertooursteeds;however,theSwisspolicewouldbenext.

TheReturn

FromtheMemoirsofDrWatsup

IthadbeenthreeyearssinceSoames’snoblesacrificehadridtheworldofMogiarty.Thedetective’spremisesat222BhadpassedtohisbrotherSpycraft,andIhadtakenupmedicalpracticeinearnest.Astoopedfigureintatteredclotheslimpedintomysurgery.“Isyouthat

doctorbloke?Whatwrotethemdee-tectivestoriesinthemmagazines?”Iacknowledgedmymedicalstatus.“Idowrite,butregrettablytheStrandhas

declinedmysubmissionstodate.”“Oh.Mustbethatothergeezer.Butyou’lldo.Igottaterriblepaininmeleg,

Doc.”“Thatwillbesciatica,”Itoldhim.“Itiscausedbyabackproblem.”“Inmeleg?”“Thenervesinyourlegaretrappedsomewhereinyourspine.”“Ohgawd!Igotnervesinmeleg?”“Lieonthecouchand—”Iobservedthedirtonhisclothes.“No,firstletme

getaclothforyoutolieon.”Iturnedmybackandopenedacupboard.“Noneedforthat,Watsup,”saidafamiliarvoice.Iturned,stared—andfainted.WhenIcameto,Soameswasbendingovermewavingsmellingsaltsunder

mynose.“Idoapologise,oldchap!Ihadpresumedyouhadlongagoworkedoutmy

cunningdeceptionandwhyithadbeennecessary.”“Notatall.Ithoughtyoudead.”“Ah.Well,yousee,whenIpushedMogiartyofftheprecipice,andsawhow

thefootprintswouldappeartoanyonelessastutethanmyself,Irealisedinaflashthatfatehadpresentedmewithagoldenopportunity.”“Yes!Isee!”Icried.“AlthoughMogiarty’shenchmenintheBritishIsleshad

beenapprehended,severalremainedatlargeonthecontinent.Iftheythoughtyoudead,youcouldweaveyourwebandtrapthem.Soyousuppliedsomemisleadingevidence,goodenoughtoconvincethebunglingSwisspolice.Sincethenyouhavespentallyourwakinghourspursuingthecriminalscumthatremained.Onebyoneyoueliminatedthem.Youtrackedthelastonedownin—

oh,Casablancaorsomeotherexoticlocale—andhewillnotbothertheworldagain.Sonowyoucanrevealthatyouarestillalive.”“Abrilliantsequenceofdeductions,Watsup.”Isilentlycongratulatedmyself.

“Thoughwronginalmosteveryparticular.”Soamesexplained:“Theonethingyougotrightwasthatthiswasaheaven-

sentopportunityformetodisappear.Butmyreasonswerenotwhatyouimagine.Ihadrunupasignificantnegativebalancebettingonthehorses,lackedthefundstoreimbursemybookmaker,andfacedthethreatofseriousbodilyharm.Havingfinallyaccumulatedthenecessaryfunds,Ipaidoffthedebtandrejoinedsociety.”Ifounditdifficulttotakein.“Iunderstandyourposition,Soames.Itcould

happentothebestofus.Buthow—?”

HowdidSoamesescapeanddisappear?Makeyourdeductionsbeforereadingon,fornarrativeimperativerequirestheimmediatepresentationoftheanswer.

TheFinalSolution

Soamessettledhimselfbesidethefire.“Ithappenedlikethis,Watsup.WhenIreachedthetopofthepath,Mogiartywaswaitingformebehindarock.Heknockedmeout,andwascarryingmetowardstheprecipicetohurlmeintotheabyss.FortunatelyIregainedconsciousnessandreachedformypistol.Intheensuingstrugglesomeshotswerefired,butnoonewashit.Mogiarty’sexertionscausedhimtoslipandfalltohisdeath.Iwasfortunatenottohavejoinedhim.”Hesaidthisinamatter-of-factvoice,asifithadbeenoflittleimport.“Intendingtosummonyou,Ilookedbackandsawasinglesetoffootprints

madebyMogiarty’sboots.Theyledfromthepathtothebrinkofthechasm,andnothingledtheotherway.IsawatoncethattheywereslightlydeeperthantheyshouldhavebeenforamanofMogiarty’sweight—acluethatIhopedyouwouldspot,Watsup,andknewthatthepolicewouldnot.Ithenwalkedbackwardstothesafetyofthepath,overlayingmyownprints,beingcarefultomakethemappearasthoughIhadcometheotherway.”“Thethoughtdidcrossmymind,Soames.ButIdismissedit,sinceIwas

unawareofyourgamblingdebts,soIcouldnotthinkofamotive.Buttheledgewasempty,thecliffimpossibletoclimb!Howdidyouconcealyourself?”Heignoredmyquestion.“IrealisedthatifIfailedtosignal,youwould

eventuallydecidethatyourpromisewasnolongervalid,andclimbthepath.Itwasbuttheworkofamomenttoclimbtotheledgeabove,wheretheoverhanghidmefromview.Therestyoucanguess.”“But—thecliffisunclimbable!”Icried.Heshookhisheadsadly.“Mydear,Watsup,Idistinctlyremembertellingyou

ofafoldinggrappling-ironthatIalwayscarryforsucheventualities.”(Seepage240.)“Youreallymustnotforgetsuchvitalinformation.Oftenonetinyfactisallittakestounlockagreatmystery.”Ihungmyhead,forIhadoverlookedthatitemofequipmentuntilthatvery

moment.Iattemptedawrygrin.“Why,Soames!Howabs—olutely,uh,ingenious!”Hegaveathinsmile,andchangedthesubject.“Acupoftea,Watsup?”“Thatwouldbedelightful,Soames.”“ThenIshallaskMrsSoapsuds—”

Thedoorswungopenandourlandlady’sheadpokedaroundit.“CanIbeofservice,MrSoames?”“—tomakeusapot,”sighedthedetective.

TheMysteriesDemystified

or,ifnot,castinanewlightbysundryextractsfromDrJohnWatsup’sextensivearchiveofcasenotes,presscuttings,andSoamesianmemorabilia;withoccasionalcontributionsfromothersources

TheScandaloftheStolenSovereign

Magnifyingglassinhand,SoamesinspectedeveryinchoftheGlitz’skitchensandaccounts.Hehadthecarpetsliftedtoseewhatwasbeneath—aremarkablecollectionbutnotofrelevancetothistale—andsearchedManuel’scrampedlodgingsintheattic.Hesampledthecontentsofseveralbottlesinthebar.Infact,hehadarrivedathisconclusionsbeforeHisLordshiphadfinisheddescribingthefactsofthecase,butitwouldneverdotomaketheprocesslookeasy,andtheopportunityofafreemaltwhiskeyshouldnotbedeclinedwithoutgoodreason.TheowneroftheGlitzHotelwaswaitinginasumptuouslyfurnishedprivate

room,pacingthefloorandglaring.“Haveyourecoveredmystolensovereign,Soames?”“No,myLord.”“Pah!IknewIshouldhavetriedMrSher—”“Ihavenotrecovereditbecausetherewasnostolensovereign.Itwasnever

missinginthefirstplace.”“But£27plus£2donotmake£30!”“Iagree.Butthereisnoreasonwhytheyshould.Thesumsaddupifyoudo

themcorrectly.”AndSoameswrote:

“Thesumof£30isnolongertheissue,”saidSoames.“Itwas,afterall,thewrongbill.Themenhavenowpaid£27,MyLord,andweshouldsubtract£2togetthe£25owedtothehotel.Notaddit.”“But—”“Youroriginalcalculationappearstomakesensebecausethenumbers29and

30aresoclosetogether.Butsuppose,forexample,thatthebillhadactuallybeen£5,sothewaiterwasgiven£25toreturn,keeping£1asatipandgivingthemeach£8.Nowthemenhadeachpaid£2,atotalof£6.Manuelhasretainedonly

£1.Thetotalofthesetwoamountsis£7.Youwouldnowask:wherehastheother£23gone?Buttheactualbillwas£5,andthehotelhasbeenpaidexactlythat.Sohowcan£23bemissingfromthehotel’sshare?Ithasbeensharedbythethreecustomers,whohavegivenasmallpartofittoManuel.”Humphshaw-Smatteringturnedpink.“Humph,”hesaid.“Pshaw.”Hepulled

himselftogether.“Yourfee,sir?”“Twenty-ninesovereigns,”saidSoames,notmissingabeat.

NumberCuriosity

1001100001100000011000000001100000000001100000000000000001

Ialsoaskedwhyitworks.That’saharderquestionbecauseyouhavetothinkandnotjustcalculate.Insteadofaformalproof,let’sjustlookatatypicalcase:11×909091.First,rewriteitintheoppositeorderas909091×11.Thisis909091×10+909091×1;thatis,9090910+909091.Addthemlikethis:

Whatnext?Startingfromtheright,0+1=1,soweget

Then1+9=0carry1:

Nowwehavetoaddthecarrytothe9and0,whichagaingives0carry1.Thisleadstoacascadeofcarries,eachconvertinga9to0carry1,untilwegetto

Finallyonlythecarrydigitisleft,leadingtotheanswer

TrackPosition

Mazesolution

ForfurtherinformationseeR.Penrose,Railwaymazes,inALifetimeofPuzzles,(eds.E.D.Demaine,M.L.Demaine,T.Rodgers),A.K.Peters,WellesleyMA2008,133–148.PicturesoftheLuppittMillenniumMonumentcanbefoundat:

http://puzzlemuseum.com/luppitt/lmb02.htm

SoamesMeetsWatsup

“Adecimalpoint?”Watsuphazarded.“No,youaskedforawholenumber.”Hepaused,struckbyasuddenrealisation.“Didyoutellmethatthesymbolmustgobetweenthetwodigits,MrSoames?”“No.”“Didyouinsistthatthedigitsbeseparatedbyaspace?”“Mydrawingwasperhapsambiguous,butIdidnotspecifyaspace.”“Ithoughtso.Wouldthismeetyourconditions?”AndWatsupwrote:

“Whichequals7.”

GeomagicSquares

Howtoassemblethejigsawsforrows,columns,anddiagonals

WhatShapeisanOrangePeel?

LaurentBartholdiandAndréHenriques.OrangepeelsandFresnelintegrals,MathematicalIntelligencer34No.4(2012)1–3.Youcandownloadasimilararticlefromarxiv.org/abs/1202.3033.

HowtoWintheLottery?

No.Thestatementsmadeareallcorrect,butthedeductionisfallacious.Toseewhy,considerthelotterythatrunseveryweekinthelittleknown

provinceofLilliputia.Herethereareonlythreeballs—1,2,3—andtwoofthemaredrawn.Youwinbygettingthosetworight.Therearethreepossibledraws:

121323

andtheseareallequallylikely.Thefirstnumberis1withprobability2/3,morelikelythan2withprobability

1/3or3withprobability0.Thesecondnumberis3withprobability2/3,morelikelythan2with

probability1/3or1withprobability0.So,bythesameargument,puntersshouldchoose13tomaximisetheir

chances.However,eachofthethreepossibilitiesisequallylikely,sothisisrubbish.Ingeneral,1ismorelikelytobethesmallestnumberinthedrawbecausein

thiscasetherearemorelargernumbersthanthereareforanyotherchoice.Notbecause1ismorelikelytobedrawn.Thesameeffectappliestotheotherpositions,butnotasobviously.

TheGreenSocksCaperIncident

“FrommydeepknowledgeofLondon’slowlife,itisimmediatelyobviouswhotheculpritis,”Soamesannounced.“Who?”“Thatisofnoconcernuntilwehaveformalproofofhisguilt,Watsup.

NothinglesswillconvinceInspectorRouladeoftheMetropolitanPolicewhenwepresenthimwithourconclusions.First,wemustlistthepossiblewaystodistributethecoloursamongthegarments.”“Icandothat,”saidWatsup.“Ihavesomesmallcommandofelementary

combinatorics.Ithasprovedusefulwhendecidingwhichlimbtoamputatefirst.”Andhewrote:

BGWBWGGBWGWBWBGWGB

“Thelettersdenotethecoloursofthegarments,intheorderjacket,trousers,socks,”Watsupexplained.“Nocolourrepeats,becauseofthewitnessreports,sotheonlypossibilitiesarethesixpermutationsofthethreeletters.”“Verygood,”saidSoames.“Andwhatshouldournextstepbe?”“Uh—totabulateallofthewaystodistributethegarmentsamongthethree

men.Thatwilltakesometime,Soames,becausethereare…uh,6×5×4…120combinations.”“Notso,Watsup.Withalittlethoughtwecaneliminatemostofthematthe

outset.Letusbeginbyfocusingonjustoneofthesuspects—GeorgeGreen,say.Suppose,forthesakeofargument,thatGreenwearsagreenjacket,browntrousers,andwhitesocks:caseGBW.”“Ah,butdoeshe?”“SoIhypothesise,forthesakeofargument.Ifthatiscorrect,itfollowsthat

theothertwosuspectscannotwearagreenjacket,orbrowntrousers,orwhitesocks,sinceonlyoneofeachtypeofgarmenthasagivencolour.SoforthosemenwecaneliminateGWB,BGW,andWBGfromthefivepossibilitiesremaining.ThatleavesonlyBWGandWGB.Which,youobserve,arecyclicpermutationsofGBW.WecanassignthesechoicestoBillBrownandWallyWhiteinonlytwoways.”Soamesbegantocompilehistable:

“ButSoames,”criedWatsup,“perhapsGeorgeGreendoesnotwearthegarmentsGBW!”“Quitepossibly,”saidSoames,unperturbed.“Thesearemerelythetoptwo

rowsofmytable.IcanmakesimilardeductionsfortheotherfivepossiblelistsforGeorgeGreen.And,ofcourse,againthepermutationsarecyclic.Therearethustwelvepossibilitiesaltogether.”Watsupcopiedouttheresultingtable:

Whenhehadfinished,Soamesnodded.“Andnow,mydearWatsup,allthatremainsistousetheevidencetoeliminatetheimpossiblecombinations—”“Becausethen,whateverremains,howeverimprobable,mustbetrue!”

Watsupcried.“Icouldnothaveputitbettermyself.Thoughinthiscase,themost

improbablefeatureisthatonlyoneofthesevillainswasinvolved.Iwouldhaveexpectedaconspiracy.“Anyway,ConstableWuggins—anadmirablefellow,Watsup,whomakesup

inperseveranceforwhathelacksinimagination—statedthatBrown’ssockswerethesamecolourasWhite’sjacket.ThatmeansthatBrown’stripleoflettersmustendwiththesameletterthatbeginsWhite’striple.Thateliminatesrows1,3,5,7,9,11,andreducesthetableto

“Next,IdeterminewhichcombinationssatisfythegoodConstable’ssecondcondition:thatthepersonwhosenamewasthecolourofWhite’strousersworesockswhosecolourwasthenameofthepersonwearingawhitejacket.Thisispurelyamatterofkeepingaclearhead.Forexample,inrow2White’strousersarewhite,sothepersonwhosenameisthecolourofWhite’strousersisWhitehimself.Hissocksaregreen.DoesGreenwearawhitejacket?No,hisjacketisgreen.Soweeliminaterow2.”“I’mnotsureIquite—”“Oh,verywell,letmedrawupanothertable!”AndSoameswrote:

“Onlyrows4,and12remain.Whichfurtherreducestheoriginaltableto

“Finally,ConstableWugginstellsusthatthecolourofGreen’ssockswasdifferentfromthenameofthepersonwearingthesamecolourtrousersasthejacketwornbythepersonwhosenamewasthecolourofBrown’ssocks.”

“Thateliminatesrow12,leavingonlyrow4.“Sonowitremainsonlytoseewhowaswearingthegreensocksinrow4.As

Isuspectedfromthestart,itwasWallyWhitewiththeentryWBG.”

ConsecutiveCubes

233+243+253=12,167+13,824+15,625=41,616=2042

Thiscanbefoundbytryingnumbersinturn.Amoresystematicmethodistoletthemiddlenumberbenandobservethat(n–1)3+n3+(n+1)3=3n3+6n=m2forsomem.Som2=3n(n2+2).Theterms3,n,n2=2havenocommonfactorsasidefromperhaps2and3.Thereforeanyprimefactorgreaterthan3mustoccurtoanevenpower(perhaps0)inbothnandn2+2.Thefirsttwonumberstosurvivethistestare4and24,and24providesasolutionbut4doesnot.

AdonisAsteroidMousterian

Thenumbersshouldbeassignedlikethis:

Order3:A=0,D=3,I=2,N=0,O=1,S=6.Order4:A=0,D=12,E=1,I=2,O=3,R=8,S=0,T=4.Order5:A=0,E=1,I=2,M=0,N=5,O=3,R=10,S=15,T=20,U=4.

Thesquaresbecome

Changingletterstonumbersandadding

Formoremagicwordsquaresandsimilarconstructions,see:JeremiahFarrell,Magicsquaremagic,WordWays33(2012)83–92.

Availableat:

http://digitalcommons.butler.edu/wordways/vol33/iss2/2

TwoSquareQuickies

1923187456,thesquareof30384.

Sincewewantthelargestnumberofitstype,it’safairbetthattheanswerstartswith9,sothisreallyhastobetriedfirst,evenifitturnsouttobefalse.Soitmustliebetween912345678and987654321,bearinginmindthatalldigitsaredifferentandthereisno0.Thesquarerootsoftheseare30205.06and31426.96.Soallwehavetodoissquarethenumbersbetween30206and31426andseewhichgiveallninenonzerodigits.Thereare1221suchnumbers.Workingbackwardsfrom31426weeventuallygetto30384.Nowthatwe’vefoundasolutionstartingwith9,wedon’tneedtoworryaboutstartingwith8orlower.2139854276,thesquareof11826.

Thewaytofindthisissimilar.

TheAdventureoftheCardboardBoxes

1Theboxeshavedimensions6×6×1and9×2×2.

Supposethedimensionsoftheboxesarex,y,zandX,Y,Z.TheirvolumesarexyzandXYZ.Thelengthofribbonis4(x+y+z)and4(X+Y+Z).Dividingoutthefactorof4,wemustsolve

xyz=XYZx+y+z=X+Y=Z

innonzerowholenumbers.Thatis,findtwotriplesofnumberswiththesameproductandthesamesum.Thesmallestsolutionis(x,y,z)=(6,6,1)and(X,Y,Z)=(9,2,2).Theproductis36andthesumis13.2Thesmallestsolutionforthreeboxesis(20,15,4),(24,10,5),and(25,8,6).Nowtheproductis1,200andthesumis39.

Inpassing,wecananswerathirdquestion,whichdidnotfeatureinSoames’sinvestigation:3Supposethattheribbonsaretiedinthemanneroftheleft-handpicture,withxbeingwidth,ydepth,andzheight.Thentheequationsbecome

xyz=XYZx+y+2z=X+Y+2Z

Ifwereplacex,y,zbyx,y,2z,andsimilarlyforX,Y,andZ,weareagainseekingtwo triples of numbers with the same product (now 2xyz = 2XYZ) and sum.However,zandZmustbeeven.Thisisthecaseinsolution(1)ifwearrangethesidesintherightorder,anditleadstothesmallestsolution(6,3,1)and(9,2,1).MyattentionwasdrawntothisproblembyMoloyDefromKolkata,India,

whohasalsofoundthesmallestsetsoffour,five,andsixwholenumberswiththesamesumandproduct:

Fourpackages

(54,50,14)(63,40,15)(70,30,18)(72,25,21)

sum=118,product=37,800.

Fivepackages

(90,84,11)(110,63,12)(126,44,15)(132,35,18)(135,28,22)sum=185,product=83,160.

Sixpackages

(196,180,24)(245,128,27)(252,120,28)(270,98,32)(280,84,36)(288,70,42)sum=400,product=846,720.

TheRATSSequence

Thenexttermis1345.Theruleis:‘Reverse,Add,ThenSort’.RATS.By‘sort’Imeanrearrangeinto

ascendingorder.Anyzerosareomitted.Forexample:

16+61=77alreadyinnumericalorder77+77=154,reorderas145145+541=686,rearrangeas668668+866=1534,rearrangeas1345

JohnHortonConwayhasconjecturedthatwhichevernumberyoustartwith,eventuallythesequenceeithergoesroundandroundsomerepeatingcycleorgetsintotheever-increasingsequence

wherethenindicatesnotapowerbutnidenticaldigitsrepeated.

MathematicalDates

Thenexttriplepalindromedaywillbeon21:1221/122112.Thenextpalindromewas20:0230/032002(Britishsystem).

TheHoundoftheBasketballs

“Indeed,madam,DrWatsupisquiteright,”Soamesconfirmed.“Therealisationthatonlyfourballsweremovedmakestherequiredarrangementobvious.”“Whatisit?”“That,madam,isinformationwhich,accordingtoyourownstatements,

shouldonlybedisclosedtotheseniorextantmemberofthemaleline.”“Namely,LordEdmundBasquet,”Ipointedout.“Whoisinacoma.Which

posesaconsiderablediff—”“Rubbish!”saidLadyHyacinth.“Youcantellme.”Itwasevidentfromher

facethatnothingwoulddeterherfromthiscourseofaction.“Verywell,”saidSoames,sketchingrapidly.“Thepood—er,giantslavering

hound—musthavemovedthefourbasketballsshowninwhitetothepositionsshowninblack.Oreitherofthetworotationsofthissolution.Butyousaidthattheorientationoftheconfigurationdoesnotmatter.”NowIunderstoodthepointofhisobscurequeryearlier.

Theoriginalconfiguration

“Wonderful!”saidLadyHyacinth.“IwillinstructWillikinstomakeitso.”“Butwillthatnotinfringetheconditionsoftheceremony?”Ienquired.“Ofcourse,DrWatsup.Butthereisnorationalreasontofearanyadverse

consequences.Thatancienttabooislittlemorethanaloadofold,er,superstition.”Amonthlater,SoameshandedmetheManchesterGarble.

“GreatScott!”Icried.“LordBasquethasdiedandBasquetHallhasburnedtotheground!Thefamily’sinsurancecompanyhasdeclinedpaymentbecauseMalevolentForcesofPureEvilareexcluded,andthefamilyisnowruined!LadyHyacinthhasbeenconfinedtoanasylumfortheincurablyinsane!”Soamesnodded.“Purecoincidence,I’msure,”hesaid.“Perhapswith

hindsightIshouldhavetoldHerLadyshipaboutthepoodle.”

DigitalCubes

370,371,and407.Despitethisproblemallegedlyhavingnomathematicalsignificance,youhave

tobequitegoodatmathstofindthefoursolutions,andevenbettertoprovetherearen’tanyothers.I’llsketchoneapproach.Sincenumberswithleadingzerosareexcluded,thereareonly900

combinationstotry.Butwecancutthisdown.Thecubesofthetendigitsare0,1,8,27,64,125,216,343,512,and729.Thesumofthethreedigitsis999orless,sowecanruleoutanynumbercontainingtwo9’s,two8’s,an8anda9,andsoon.Supposeonedigitis0.Thenthenumberisasumoftwocubesfromthelist.

Ofthe55suchpairsonly343+27=370and64+343=407havetherequiredproperty.Wemaynowassumenodigitis0.Supposeoneofthemis1.Asimilar

calculationleadsto125+27+1=153and343+27+1=371.Wemaynowassumenodigitis0or1.Nowwehaveasmallerlistofcubesto

workwith.Andsoon.Shortcuts,suchasconsideringoddandevennumbers,shortenthe

calculationsfurther.It’sabitlong,butasystematicapproach(asSoamesalwaysrecommends)getstherewithoutanyseriousproblemsalongtheway.

NarcissisticNumbers

Hereweallowleadingzeros.

Clueless!

Watsup’ssolutiontocluelesspseudoku

“Soames!”Icried.“I’vesolvedit!”“Yes,themurderesswasGra¨finLiselottevonFinkelstein,ridingher

thoroughbredPrinzIgorandtowingthreecarthorsesbehindtoobscurethetracksinthe—”“No,no,Soames,notyourcase!Thepuzzle!”Hegavemyscrawledsolutionacursoryglance.“Correct.Aluckyguess,no

doubt.”“No,Soames,Ireasoneditoutusingthelogicalprinciplesthatyouhave

impresseduponmyconsciousness.First,Irealisedthatthenumbersineachregionmustsumto20.”“Becausethetotalofthenumbersinallsquaresis(1+2+3+4)×4=40,to

bedividedequallybetweenthetworegions,”Soamessaiddismissively.“Exactly.Now,onceithadoccurredtometoconcentrateonthelargerregion,

thesolutionbegantofallintoplace.Thatregionhasfourcellsinarowalongthebottom,whichmustcontain1,2,3,4insomeorder,andthoseaddto10,whatevertheordermaybe.Sotheremainingthreerowsmustalsoaddto10.Theonlywaythiscanhappenisifthetoprowcontains1,2,3insomeorder;thesecondrowcontains1,2insomeorder,andthethirdrowcontains1.”“Why?”“Anyotherchoicewouldmakethetotaltoobig.”“Youareindeedlearning,Watsup.Verygood:continue.”Ismiledatthisfaintpraise,sincegettinganypraisefromSoameswaslike

makingmarmaladefromtheIsleofWight.“Well—it’snoweasytoverifythatthereisonlyonepossiblewaytocompletethearrangement.Thenumbersinthe

secondpieceareforced:forexample,thetoprowmustendin4,andthentheother4’smustgodownthediagonal;thenthetwo3’sareforced,andfinallythe2goesintheremainingposition.”ThispuzzlewasinventedbyGerardButters,FrederickHenle,JamesHenle,

andColleenMcGaughey,Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.Seealsothewebsite:

http://www.math.smith.edu/~jhenle/clueless/

ABriefHistoryofSudoku

ThetwobasicallydifferentsolutionstoOzanam’spuzzleare:

Remember:eachofthesegivesriseto576solutionsbypermutingthevaluesandsuits,sodon’tbesurprisedifyoursolutionslookdifferent.Ifyoustartwiththetoprow (orrearrangeyoursolutionintothisform)youneedonlythinkaboutpermutingtheotherthreerows.

Once,Twice,Thrice

TheCaseoftheFace-DownAces

“Itisalltrickery,Watsup.Withtherightpreparation,thetrickworksautomatically,nomatterwhichsequenceoffoldstheaudiencechooses.”“Dashedclever,what?”saidI.Soamesgrunted.“WhenWhodunnipreparedthepack,heplacedthefouraces

inpositions1,6,11,and16fromthetopdown.So,whenthepackwasdealtintoasquare,theaceslayalongthediagonalfromtoplefttobottomright.Buttheywerefacedown,soofcourseyouwouldnothavebeenawareofthedeception.“Imagineturningthecardsthatliealongthediagonalfaceup.Thenthesquare

wouldhaveapatternlikeachessboard,withtheacesalongthediagonal:

Whodunni’sinitialarrangementwiththediagonalcardsturnedover

“Now,thisarrangementhasawonderfulmathematicalproperty.Howeveryoufoldthesquare,atanystagethecardsthatendupinagivenpositionwillallfacethesameway:eitheralluporalldown.”“Really?”“Letustry.Forinstance,wemightbeginbyfoldingalongthecentralvertical

line.Thinkofthetoprowofcards.Thethirdcard(up)turnsover(down)andisplacedontopofthesecondcard—alsodown.Andthefourthcard(down)turnsover(up),andisplacedontopofthefirstcard—alsoup.”Ibegandimlytoseehowitworked.“Andthesamegoesfortheotherrows?”

“Indeed.Thisfirstfoldcreatesarectangle,madefromcardsorsmallpilesofcards.Thecardsineachpileallfacethesameway(upordown),andthesetofpileshasthesamechessboardpatternofupsanddownsastheoriginalsetofcards.Sothesamethinghappensforthenextfold,andthenext.Bythetimewereachasinglepile,allofthecardsinthepilefacethesameway.”“Yes,but—whenwestarted,thecardsonthediagonalwerethewrongwayup

comparedtothechessboardpattern,”saidI.Itwasintendedasanobjection,buthebeamedatmyinsight.“Exactly!So,

afterfolding,theywillagainbethewrongwayup.Soinsteadofapileofsixteencardsallfacingthesameway,youwillhaveapilewithtwelvecardsfacingoneway,andthefouracesfacingtheother.”Itwasdevilishlycunning.Thechessboardpatternhaswhatmathematicianscall‘coloursymmetry’.The

foldlinesactlikemirrors,andthemirrorimageofeachcardsitsontopofacardthatfacestheotherway.Thisideaisusedtostudyhowtheatomsincrystalsarearranged.Thecunningbitistoturnthemathematicsintoaneffectivecardtrick.Whodunnididn’tdothat.Followinghisusualmodusoperandi,hestolethetrickfromitsinventorArthurBenjamin,amathematicianandmagicianatHarveyMuddCollegeinCalifornia.

JigsawParadox

Neithershapeisatriangle.Thefirstbulgesslightlyupwardsalongtheslopingedge,thesecondbulgesslightlydownwards.That’swherethemissingsquarehasgone.

TheCatflapofFear

Soamesnoddedinsatisfaction.“Ihaveit,Watsup!Cirrhosisgoesout,Dysplasiagoesout,Aneurysmgoesout,Cirrhosiscomesbackin,Borborygmusgoesout,Cirrhosisgoesout.”Webeganthedelicateprocessofenticingcatsoutthroughtheflapand

stuffingthembackin.“Takecare,Soames!”Iwhispered.“Onemistakeandthisentireareawillbeasmokingcrater.IdonotwishtopresentmyselformycatsatthePearlyGatesquiteyet.Iamwearingunpressedtrousers,andthecatsneedbrushing.”“Donotconcernyourself,Watsup,”saidhe,grabbingCirrhosisbeforethe

wretchedanimalcouldhoofitoverthefence.“Youcanhavetotalconfidenceinmysolution.”“Idonotdoubtit,Soames,”Ireplied,castinghastilyaboutforsomething

solidtohidebehind.“Er—howdidyoumakeyourdeductions?”Heborrowedmynotebookandapencil.“Thereare16possibilitiesforwhichcatsareinthehouse:ABCD,ABC,

ABD,andsoonuntilnonearepresent(callthis*).Useanarrow todenoteapossiblemove:onecatthroughtheflap.“ThefirstconditionrulesoutACandABC.ThesecondrulesoutBDand

BCD.ThethirdrulesoutAD.ThefourthrulesoutCD.ThefifthrulesoutthechangeA *.ThesixthrulesoutB *.“Now,ABCD ACDorABD.However,ACD AC,AD,orCD,allof

whichareruledout.ThereforeABC ABD.SinceABD ADandABD BDareruledout,wemusthaveABD AB.ButAB AispointlesssinceAcan’tgooutifnoothersarepresent.SoAB B.However,Bcannotthengoout,sosomeothercatmustcomebackin.ButB ABinvolvesAcomingstraightbackin,andB BDisruledout,soB BC.NowBC C *.“Thereisalsoavisualwaytoseethis,whichinsomewaysissimpler,”he

added,andsketchedadiagram.“Thispictureshowsall16possiblecombinationsofcats,withthethinlinesrepresentingpossiblechangesasacatgoesinorout.Theblackdotsareruledout,thetwocrossesruleouttwoofthelines.TheboldlineistheonlypaththatrunsfromABCDto*usingonlypermitteddotsandlines,butneverbacktracks.”

Catflapconditions

Shortlyafterwards,Iwasreunitedwithmyfurryfriends.“Soames,howcanIeverthankyou?”Icried,claspingtheanimalsjoyouslytomychest.Helookeddownathisjacket.“Bybrushingyourcatsmoreoften,Watsup.”

PancakeNumbers

1No.2Somestacksoffourpancakesneedfourflips;forexample,theonebelow.Seethefigurebelowthatfortheothertwo.Nostackneedsmorethanfourflips.

Astackthatneedsfourflips

Here’sasystematicmethodforprovingthosestatements.Thediagramshowstherequiredfinalarrangement1234atthetop,wherethesizesarelistedinorderfromthetopdown.Weworkbackwardsfromthis.Thesecondlineshowsallarrangementsthatcanbeobtainedfrom1234byoneflip.Thesearealsothearrangementsthatcanreach1234inoneflip.(Thesameflipdonetwiceputseverythingbackwhereitwas.)Thethirdlineshowsallarrangementsthatcanbe

obtainedfromthefirstlinebyoneflip.Thesearealsothearrangementsthatcanreach1234intwoflips.Noticethatpreciselyoneentryinthethirdrowcanbereachedfromtwointhesecond,namely1324.Sothestructureofthediagramlooksslightlyasymmetricthere.Rows1,2,3contain21ofthe24possiblestackorders.Themissingonesare

2413,3142,and4231.Row4showshowthesecanbeobtainedfromrow3byonemoreflip—or,reversingtheseriesofflips,howtoconvertthemto1234infourflips.(Theotherlinkstorow4areomittedsincetheymakethediagrammorecomplicatedandwedon’tneedthem.)Thefigureaboveinanswer2isthearrangement2413convertedintostacks.

Stacksthatneed1,2,3,or4flipstoputtheminorder

3Eitherthebiggestpancakeisatthetop,orit’snot.Ifnot,insertthespatulaunderitandflip.Nowit’satthetop.Insertthespatulaatthebottomofthestackandfliptheentirething.Nowthebiggestisatthebottom.Soatmosttwoflipswillputthebiggestpancakeatthebottom.Leaveitthereandrepeatforthenextbiggestpancake:atmosttwoflipswillplaceitdirectlyontopofthebiggest.Repeatforthenextbiggest,andsoon.Ittakesatmosttwoflipstogeteachsuccessivepancakeinitscorrectposition,soatmost2nflipswillachievethisfortheentirestackofnpancakes.4P1=0,P2=1,P3=3,P4=4,P5=5.Thepancake-sortingproblemgoesbacktoJacobGoodmanin1975,who

publisheditunderthepseudonymHarryDweighter.(Sayitaloudandpronouncethesurnameas‘Dwayter’togetthejoke.)Thesolutionisknownforallnupto19,butnotfor20.Theresultsare:

Thepancakenumberstendtoruninsequence,goingupby1asnincreases.ForexamplePnis3,4,5,6whenn=3,4,5,6.Butthispatterngoeswrongwhenn=7becauseP7=8,not7.Afterthat,thereisajumpof2atn=11,andagainatn=19.Theupperestimateof2nflips,myanswertoquestion3,canbeimproved.In

1975WilliamGates(yes,theBillGates)andChristosPapadimitrioureplaceditby(5n+5)/3.GatesandPapadimitrioualsodiscussedtheburntpancakeproblem.Hereeach

pancakeisburntononeside,whichcouldbethetoporthebottom,andyouhavetogetalltheburntsidesonthebottomaswellasstackingthepancakesintherightorder.In1995DavidCohenprovedthattheburntpancakeproblemneedsatleast3n/2flips,andcanbesolvedwithatmost2n–2flips.Ifyou’rethinkingoftacklingn=20,bearinmindthatthereare

2,432,902,008,176,640,000

differentstackstostartfrom.

TheCaseoftheCrypticCartwheel

“Thediameterofthewheelis58inches,ofcourse,”saidSoames.“ItisanelementaryapplicationofPythagoras’stheorem.”Ithoughtaboutthis.Ihavesomesmallfacilityingeometryandalgebra.“Let

metry,Soames.Itaketheradiusofthewheeltober.Theshadedtriangleinyourdiagramisright-angled,withhypotenuserandtheothersidesbeingr–8andr–9.So,asyouhinted,IcanapplyPythagoras,getting

(r–8)2+(r–9)2=r2

Thatis,

r2+34r=145=0

Thinkofatriangle…

Istaredatthesymbols,temporarilyhalted.“Thequadraticfactorises,Watsup:

(r–29)(r–5)=0

“Soitdoes!Whichmeansthatthesolutionsarer=29andr=5.”

“Yes.Butyoumustrememberthatthediameteris2r,whichis58or10.However,thesolution10inchesisruledoutbecausethediameterismorethan20inches.Allthatremains—”“Is58inches,”Ifinishedforhim.

TheV-shapedGooseMystery

FlorianMuijresandMichaelDickinson,Birdflight:Flywithalittleflapfromyourfriends,Nature505(16January2014)295–296.StevenJ.Portugalandothers,Upwashexploitationanddownwashavoidance

byflapphasinginibisformationflight,Nature505(16January2014)399–402.

AmazingSquares

Themainideacanbeexpressedinfullgeneralityusingalgebra,butI’llforegoformalitiesandillustrateitbyexample.Lookattheprocessinreverse,startingwith

92+52+42=82+32+72

andexpandingitto

892+452+642=682+432+872

It’seasytocheckthefirstequation,whichishoweverythinggetsstarted,butwhydoesthesecondequationhold?Theactualvalueofatwo-digitnumber[ab]is10a+b.Sowecanwritethe

left-handsideas

(10×8+9)2+(10×4+5)2+(10×6+4)2

whichis

100(82+42+62)+20(8×9+4×5+6×4)+92+52+42

Similarly,theright-handsidebecomes

100(62+42+82)+20(6×8+4×3+8×7)+82+32+72

Comparingthese,thefirsttermsareequalbecause62+42+82isjust82+42+62inadifferentorder,andthethirdtermsareequalbecausewestartedfromthose.Sowejustneedtoseewhetherthemiddletermsareequal;thatis,whether

8×9+4×5+6×4=6×8+4×3+8×7

Infact,bothare116.Everythinguptothispointwouldhaveworkedifwe’dusedanythreesingle-

digitnumbersinplaceof8,4,6.Sowejusthavetochoosethesenumberstomakethefinalexpressionsequal.Therestoftheexplanationissimilar.

TheThirty-SevenMystery

WithsomeproddingfromSoamesalongtheway,Ieventuallyrealisedthatthekeytothemysteryistheequation111=3×37.Thethree-digitnumbersthatproducelonglistsofrepeateddigitswhensubjectedtomyprocedureturnouttobethosethataremultiplesof3.Thisisthecasefor123,234,345,456,and126,forexample.Forsuchnumberstheprocedureisequivalenttomultiplyingmanyrepetitionsofasmallernumber,onethirdthesize,by3×37,whichis111.Asanexample,considerSoames’s486.Thisis3×162.Therefore,

multiplying486486486486486486by37isthesameasmultiplying162162162162162162by111.Since111=100+10+1wecandothisbyaddingtogetherthenumbers

162162162162162162001621621621621621620162162162162162162

Fromrighttoleft,weobtain0+0+2=2,then0+2+6=8.Afterthat,weget2+6+1,6+1+2,1+2+6,overandoveragain,untilwegetneattheleft-handend.Butthesearethesamethreenumbersaddedinvariousorders,sotheresultisthesameineachcase—namely9.WhenSoamesfirstexplainedthistome,Iraisedanobjection.“Yes,butwhat

ifthethreenumbersaddtomorethan9?Thenthereisacarrydigit!”Hisreplywasbriefandtothepoint.“Yes,Watsup:thesamecarrydigitevery

time.”Ieventuallyrealisedthatthismeansthatagain,onedigitwillrepeatmanytimes.“Thereare,ofcourse,moreformalproofs,”Soamesremarked,“butIthink

thisonemakestheideaclear.”Afterwhichhereturnedtohischairwithapileofnewspapersandsaidnomorethatevening,whileIwentdownstairstobegaplateofGorgonzolasandwichesfromMrsSoapsuds.[ThisitemwasinspiredbysomeobservationsmadebyStephenGledhill.]

AverageSpeed

We’reusingthewrongmean.Weshouldbeusingtheharmonicmean(explainedbelow),notthearithmeticmean.Wenormallydefinethe‘averagespeed’forsomejourneytobethetotal

distancetravelleddividedbythetotaltimetaken.Ifthejourneyisbrokenupintopieces,thentheaveragespeedforthewholetripisnot,ingeneral,thearithmeticmeanoftheaveragespeedsforthepieces.Ifthepiecesaretravelledinequaltimes,thearithmeticmeanworks,butitdoesn’tiftheycoverequaldistances,whichisthecasehere.Equaltimesfirst.Supposeacardrivesatspeedafortimet,andthenatspeed

bforthesametimet.Thetotaldistanceisat+bt,travelledintime2t.Sotheaveragespeedis(at+bt)/2t,whichequals(a+b)/2,thearithmeticmean.Next,equaldistances.Nowthecardrivesdistancedatspeeda,takingtimer.

Thenitdrivesdistancedagain,atspeedb,takingtimes.Thetotaldistanceis2d,andthetotaltimeisr+s.Toexpressthisintermsofthespeedsaandb,observethatd+ar=bs.Sor=d/aands=d/b.Theaveragespeedistherefore:

Thissimplifiesto2ab/(a+b),whichistheharmonicmeanofaandb.Itisthereciprocalofthearithmeticmeanofthereciprocalsofaandb,wherethereciprocalofxis1/x.Thisoccursbecausethetimetakenisproportionaltothereciprocalofthespeed.

FourCluelessPseudoku

Cluelesspseudokuanswers

ThesepuzzlesalsocomefromGerardButters,FrederickHenle,JamesHenle,andColleenMcGaughey.Creatingcluelesspuzzles,TheMathematicalIntelligencer33No.3(Fall2011)102–105.

ThePuzzleofthePurloinedPapers

“Charlesworthwasthethief,”saidSoames.“Areyoucertain,Hemlock?Muchhangsuponyourbeingright.”“Therecanbenodoubt,Spycraft.Theirstatementsare:Arbuthnot:Burlingtondidit.Burlington:Arbuthnotislying.Charlesworth:ItwasnotI.Dashingham:Arbuthnotdidit.

Weknowthatoneofthemenspeakstrulyandtheotherthreelie.Therearefourpossibilities.Letustrytheminturn.“IfonlyArbuthnotistellingthetruththenhisstatementinformsusthat

Burlingtonistheguiltyparty.However,Charlesworthislying,soitwasCharlesworth.Thisisalogicalcontradiction,soArbuthnotisnottellingthetruth.“IfonlyBurlingtonistellingthetruththen—”“Charlesworthislying!”Icried.“SoitwasCharlesworth!”

Soamesglaredatmeforstealinghisthunder.“Thatisso,Watsup,andtheotherstatementsareconsistentwithit.SowealreadyknowthatCharlesworthisthethief.However,itisworthcheckingtheothertwopossibilitiestoavoideventheremotepossibilityoferror.”“Absolutely,oldchap,”saidI.Hetookouthispipebutdidnotlightit.“IfonlyCharlesworthistellingthe

truththenBurlington’sstatementisfalse,soArbuthnotistellingthetruth,againacontradictionsinceheislying.“IfonlyDashinghamistellingthetruththenthesamecontradictionarises.“SotheonlypossibilityisthatBurlingtonisthesolepersontellingthetruth,

andthatconfirmsthatthethiefisCharlesworth.AsWatsupsoastutelydeduced.”“Thankyou,gentlemen,”saidSpycraft.“IknewIcouldrelyonyou.”He

gesturedandashadowyfigureenteredtheroom.Theyheldawhisperedconversation,andheleftagain.“TheCaptain’sresidencewillbesearchedforthwith,”saidSpycraft.“Iamconfidentthedocumentwillbefoundthere.”“ThenwehavesavedtheEmpire!”Ireplied.“Untilthenexttimesomeoneleavessecretdocumentsontheseatofacab,”

saidSoamesdrily.OnourwayoutIwhisperedtomycompanion:“Soames,ifSpycraftisan

expertinprimenumbers,whatonearthishedoingworkingincounter-espionage?Therecanbenopossibleconnection,canthere?”Hestaredatmeforamoment,andshookhishead.Whethertoconfirmthe

absenceofanyconnection,ortowarnmenottopursuethematter,Idonotknow.

AnotherNumberCuriosity

123456×8+6=9876541234567×8+7=987654312345678×8+8=98765432123456789×8+9=987654321

It’snottotallyclearwhat‘ought’tocomenext:perhaps

1234567890×8+10

whichis9876543130,ormaybeIshouldhavereplacedthat0by10anddone

somecarryinguntilitallsettlesdown,giving

1234567900×8+10

whichis98765432730.Ithinkwecanagreethat,eitherway,thepatternstopshere.

ProgressonPrimeGaps

TheElliott-HalberstamConjecture[PeterElliottandHeiniHalberstam,Aconjectureinprimenumbertheory,SymposiaMathematica4(1968)59-72]isverytechnical.Writeπ(x)forthenumberofprimeslessthanorequaltox.Foranypositiveintegerqandahavingnofactor(otherthan1)incommonwithq,letπ(x;q,a)bethenumberofprimeslessthanorequaltoxthatarecongruenttoa(modq).Thisisapproximatelyequaltoπ(x)/ (q)where isEuler’stotientfunction,thenumberofintegersbetween1andq–1thathavenofactorincommonwithq.Considerthelargesterror

TheElliott-HalberstamConjecturetellsushowbigthiserroris:itstatesthatforallθ<1andA>0thereexistsaconstantC>0suchthat

forallx>2.Itisknowntobetrueforθ< .

TheSignofOne:PartTwo

Here’sonesolution:

SeeTheSignofOne:PartThree,page115,foranexplanation.

Euclid’sDoodle

Youcoulddoitbyhandusingprimefactors,givenadayortwo.You’dhavetoworkoutthat

44,758,272,401=17×17,683×148,89113,164,197,765=5×17,683×148,891

Thenyou’dconcludethatthehcfis17,683×148,891,whichequals2,632,839,553.UsingEuclid’salgorithm,thewholecalculationgoeslikethis:

(13,164,197,765;44,758,272,401) (13,164,197,765;31,594,074,636)(13,164,197,765;18,429,876,871) (5,265,679,106,

13,164,197,765)(5,265,679,106;7,898,518,659) (2,632,839,553;5,265,679,106)(2,632,839,553;2,632,839,553) (0;2,632,839,553)

Sothehcfis2,632,839,553.

123456789TimesX

12345679×1=12345678912345679×2=246913578

12345679×3=37037036712345679×4=49382715612345679×5=61728394512345679×6=74074073412345679×7=86419752312345679×8=98765431212345679×9=1111111101

Thesemultipleshaveallninenonzerodigitsinsomeorder,exceptwhenwemultiplybysomethingdivisibleby3(thatis,3,6,and9).

TheSignofOne:PartThree

Since

wecanusetherepresentationof7withtwo1’sfrompage279toget62withfour1’s.ForalongtimeSoamesandWatsupdespairedofgetting138withfour1’s,

butbyusingWatsup’sinsightaboutsquarerootsandfactorials,andbeingsystematic,theyeventuallydiscoveredthatitispossibletoget138usingonlythree1’s.Againthestartingpointis7writtenwithtwo1’s,andthen

andfinally

whichisacleverwaytomultiplyby3usingjustoneextra1.

TossingaFairCoinIsn’tFair

PersiDiaconis,SusanHolmes,andRichardMontgomery,Dynamicalbiasinthecointoss,SIAMReview49(2007)211–223.Foranon-technicalsummaryseePersiDiaconis,SusanHolmes,andRichard

Montgomery,Thefifty-onepercentsolution,What’sHappeningintheMathematicalSciences7(2009)33–45.Similareffectsoccurfordice—infact,notjustfortheusualcubebutforany

regularpolyhedron.SeeJ.Strzalko,J.Grabski,A.Stefanski,andT.Kapitaniak,Canthedicebefairbydynamics?InternationalJournalofBifurcationand

Chaos20No.4(April2010)1175–1184.

EliminatingtheImpossible

“Youromission,”saidSoames,“wastofailtoobservethatthewinemaymove,aswellastheglasses.Imerelypickupthesecondandfourthglasses,andpourtheircontentsintotheseventhandninth.”

MusselPower

MoniquedeJager,FranzJ.Weissing,PeterM.J.Herman,BartA.Nolet,andJohanvandeKoppel.Lévywalksevolvethroughinteractionbetweenmovementandenvironmentalcomplexity,Science332(4June2011)1551–1553.

ProofThattheWorldisRound

Onpage74wesawthatwhencalculatingaveragespeedsoverafixeddistanceweshouldusetheharmonicmean,notthearithmeticmean.Theharmonicmeanalsoturnsupintheestimationofthedistancebetweentwoairportswhenthewindspeedistakenintoaccount,forasimilarbutslightlydifferentreason.Togetasimplemodel,assumethattheaircraft’sspeedrelativetotheairisc,itspathisastraightlineandthewindblowsalongthatlineinafixeddirectionwithspeedw.Assumebothcandwareconstant.Thena=c–wandb=c+w,andwewanttoestimatedfromthetimesrands.Togetridofw,wefirstsolveforaandb,gettinga=d/randb=d/s.Therefore

c–w=d/rc+w=d/s

Adding,2c=d(1/r+1/s).Soc=d(1/r+1/s)/2.Iftherehadbeennowind,asingletripwouldhavetakentimet,whered=ct.Therefore

t=d/c=d/[d(1/r+1/s)/2]=1/[(1/r+1/s)/2]

whichistheharmonicmeanofrands.Inshort:ifweareworkinginunitsofaircraft-hours,thenthissimplemodelof

theeffectofwindimpliesthatweshouldusetheharmonicmeanofthetraveltimesinthetwodirections.

123456789TimesXContinued

12345679×10=123456789012345679×11=135802467912345679×12=148148146812345679×13=160493825712345679×14=172839504612345679×15=185185183512345679×16=197530862412345679×17=209876541312345679×18=222222220212345679×19=2345678991

Thesemultipleshavealltendigits0–9insomeorder,exceptwhenwemultiplybysomethingdivisibleby3…Untilwegetto19,whenthepatternstops(19isnotamultipleof3:answerhastwo9’sandno0).Butitpicksupagain:

12345679×20=246913578012345679×21=2592592569

(21isamultipleof3,sorepetitionsOK)12345679×22=271604935812345679×23=2839506147

Thenextexceptionsoccurat28and29.Itworksfor30–36,andthenfailsfor37.Istoppedcalculatingatthatpoint.What’sgoingonhere?Ihavenoidea.

TheRiddleoftheGoldenRhombus

Soamesfinishedtighteningtheknot,flattenedit,andheldituptothelight.“Why,itisapentagon!”Icried.“Moreprecisely,Watsup,itappearstobearegularpentagonwithone

diagonalvisibleandthreemorehidden.Observetheabsenceofadiagonallinerunninghorizontally.Werethattobeadded,forinstancebyfoldingthestriponemoretime,wewouldobserve—”

Theflattenedknot(dottedlinesshowhiddenedges)

“Afive-pointedstar!Apentacle!Usedinblackmagictoconjuredemons!”Soamesnodded.“Butwithoutthatfinalfold,andthereforehavingoneedge

missing,thepentacleisincompleteandthedemonwillescape.Sothesymbolrepresentsathreattounleashdemonicforcesupontheworld.”Hegaveahumourlesssmile.“Ofcourse,therearenodemonsinasupernaturalsense,andtheycanneitherbeconjurednorunleashed.Buttherearehumansofademonicdisposition—”“SuchastheAl-Jebraterroristorganisation!”Icried.“Theyhavepursuedme

fromAl-Jebraistanwithweaponsofmathsinstruction!”“Calmyourself,Watsup.No,theorganisationIhaveinmindisthe

MathemagicalAssociationofNumerica.Itisanobscuregroup,whichIstronglysuspecttobeafrontforoneofMogiarty’sdevilishschemes.Ihaveencountereditbefore,andInowhavethefinallinkinthechaintostrikeablowagainstthefoulProfessoranddestroythispartofhisworldwidewebofcriminalityforever.Provided…““Providedwhat,Soames?”

“ProvidedthatwecanofferincontrovertibleproofwhenthecasecomestoCourt.Howdoweknowthatthepentagonisregular?”“Isthatnotabsurdlysimple?”“Onthecontrary,youwillshortlybeassuringmethatitisincrediblysubtle

andpossiblyfalse—thoughinpointoffactthetrueansweriswhatonewouldnaivelyguess.Idaretosurmisethatonceweestablishthatfact,allelsewillfollow,buttheknot’sappearancetotheeyeisnotenough.Ishall,however,assumethatthearrangementoflinesinthefigureiscorrect,sowehaveapentagonwithfourofitsdiagonals.Isittrulyregular?Thatremainstobeseen.Iftrue,itmustbeaconsequenceoftheconstantwidthofthestrip.“Letus,then,labelthecornersinthemannerofthegreatEuclidof

Alexandria,andpursueourgeometricaldeductions.”

Theflattenedknot,labelled.LineCDisomittedbecausewedonotyetknowwhetheritisparalleltoBE.

ImustwarnmyreadernowthattheremainderofthediscussionwillappealonlytothosewithsomefacilityinEuclideangeometry.“Ibegin,”saidSoames,“withafewsimpleobservations.Theycanbeproved

withoutgreatdifficultyusingbasicgeometry,soIomitthedetails.“First,noticethatiftwostripswithparalleledges,havingthesamewidth,

overlap,thentheirintersectionisarhombus—aparallelogramwithallfoursidesequal.Moreover,iftwosuchrhombuseshavethesamewidthandthesameside,theyarecongruent—theyhaveexactlythesamesizeandshape.Thediagramoftheflattenedknotthereforeincludesthreemutuallycongruentrhombuses.”“Whyonlythree?”Iasked,puzzled.“BecauseCDandBEdonotcoincidewithedgesofthestrip,sowecannotyet

saythesameofCDRBorDESC.ThisiswhyIhavenotdrawnlineCD.”Ihadnotnoticedthat.“Itisincrediblysubtle,then,Soames.Infact,itmight

evenbefalse!”

Threecongruentrhombusesintheflattenedknot

Hesighed,Iknownotwhy.“Nowwecometothecentralpointinmydeductions.Thediagonalsofarhombusbisectitsangles,andoppositeanglesareequal.”SoamesmarkedfouroftheangleswiththeGreekletterθ(theta),seetheleft-handfigure.

Left:Fourequalangles.Right:Fivemoreangles,allequaltothefirstfour(ingrey).

“Forsimilarreasons,angleCABisalsoequaltoθ.SincerhombusesDEATandPEABarecongruent,Icanmarkfourmoreangleswithθ.Thisleadstotheright-handfigure.“Now,Watsup:whatinstantlyspringstomind?”“Thereareanawfullotofθ’s,”Irepliedatonce.HegrimacedandIheardalowgrowlinhisthroat,Iknownotwhy.“Itisas

plainastheneckonaverytallgiraffe,Watsup!ConsidertriangleEAB.”Iconsideredit,initiallywithoutenlightenment.Well…thetrianglealsohada

lotofθs.Infact…allofitsangleswerecomposedofθ’s!NowIsawit.“Theanglesofatriangleaddto180°,Soames.Inthistriangle,theanglesareθ,θ,and3θ.Sotheirsum5θisequalto180°,whenceθ=36°.”“Wewillmakeageometerofyouyet,”saidhe.“Nowtherestoftheproofis

easy.ThelinesDE,EA,AB,andBCareofequallength,beingsidesofcongruentrhombuses.Theangles∠DEA,∠EAB,and∠ABCareallequal,sincetheyoccurincongruentrhombuses,andoneofthemis∠EAB,equalto3θ,whichis108°.Thereforeallthreeanglesare108°.Butthisistheinteriorangleofaregularpentagon.”“ThereforeDEABCarethecornersofaregularpentagon,andIcancomplete

thefigurebydrawingsideCD!”Icried.“Howabsur—”Icaughthiseye.“Er,elegant,Soames!”

Heshrugged.“Atrifle,Watsup.EnoughtowrecktheMathemagicalAssociationofNumericaandcauseMogiartysomemomentaryannoyance.Themanhimself…Ifearhewillproveafarhardernuttocrack.”

WhydoGuinnessBubblesgoDownwards?

E.S.Benilov,C.P.Cummins,andW.T.Lee.WhydobubblesinGuinnesssink?arXiv:1205.5233[physics.flu-dyn].

TheDogsThatFightinthePark

“Thedogstook10secondstocollide,”Soamesdeclared.“Ishalltakeyourwordforit,”saidI.“But,merelytosatisfymycuriosity,

howdidyouarriveatthatfigure?”“Theproblemissymmetric,Watsup,andsymmetryoftensimplifies

reasoning.Thethreedogsarealwaysatthecornersofanequilateraltriangle.Thisrotatesandshrinks,butkeepsitsshape.Therefore,fromtheviewpointofoneofthedogs—sayA—itisalwaysrunningstraighttowardsthenextdogB.”“Doesnotthetrianglerotate,Soames?”“Indeeditdoes,butthatisirrelevant,forwemayperformthecalculationina

rotatingframeofreference.Whatmattersishowfastthetriangleshrinks.DogBisalwaysrunningat60°tothelineAB,becausethedogsalwaysformanequilateraltriangle.SothecomponentofitsspeedinthedirectionofdogAis1/2×4=2yardspersecond.ThereforeAandBareapproachingeachotheratacombinedspeedof4+2=6yardspersecond,andcovertheinitialseparationof60yardsin60/6=10seconds.”

WhatdogBdoesintheframeofreferenceofdogA

WhyDoMyFriendsHaveMoreFriendsThanIDo?

Supposethesocialnetworkhasnpeople,andpersonihasxifriends.Thentheaveragenumberoffriends,overallmembers,is

Tothinkaboutcolumn3inthetable,theweightedaverageofhowmanyfriendseachfriendjofpersonihas,weuseastandardmathematicaltrickandworkonpersonjinstead.Theyturnupasafriendtoxjpeople—namely,theirownfriends—andtheycontributexjtothetotalforeachofthosefriends.Sothecasesforwhichpersonjoccursasafriendcontributex2jtothetotal.Thenumberofentriesincolumn3isx+1+…+xn.Sotheweightedaverageofhowmanyfriendseachfriendhasis

Iclaimthatforanychoicesofthexjwealwayshaveb>a,unlessallthexjareequal,inwhichcaseb=a.Thisfollowsfromastandardinequalityrelatingtheaveragetowhatengineerscallthe‘rootmeansquare’(squarerootoftheaverageofthesquares):

withequalityonlywhenallxjareequal.Squaringthisandrearranging,wegeta<bexceptwhenallxjareequal,asrequired.Forfurtherinformation,see:

https://www.artofproblemsolving.com/Wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality

TheAdventureoftheSixGuests

Soames’sremarkisanexampleofRamseyTheory,abranchofcombinatoricsnamedafterFrankRamsey,whoprovedamoregeneraltheoremofasimilarkindin1930.HisbrotherMichaelbecameArchbishopofCanterbury.Let’sworkuptoitgently.Supposethatanumberofpeopleareseatedroundatable,witheveryonebeingconnectedtoeveryoneelsebyeitheraforkoraknife.Pickanytwonumbersfandk.ThenthereissomenumberR,dependingonfandk,suchthatifatleastRpeoplearepresenttheneitherfofthemarejoinedbyforks,orkbyknives.ThesmallestsuchRisdenotedbyR(f,k)andcalledtheRamseynumber.

Soames’sproofshowsthatR(3,3)=6.Ramseynumbersareextraordinarilydifficulttocalculate,exceptinafewsimplecases.Forexample,itisknownthatR(5,5)liesbetween43and49,buttheexactvalueremainsamystery.Ramseyprovedamoregeneraltheoreminwhichthenumberoftypesof

connection(knife,fork,whatever—coloursareamorecommonimage,butSoamesworkswithwhateveristohand)canbeanyfinitenumber.Theonlyknownnon-trivialRamseynumberformorethantwotypesofconnectionisR(3,3,3),whichis17.Thereareinnumerablegeneralisationsofthisidea.Inveryfewcasesisthe

exactnumberconcernedknown.Thepaperthatstarteditalloffis:F.P.Ramsey,Onaproblemofformallogic,ProceedingsoftheLondonMathematicalSociety30(1930)264–286.Asthetitlesuggests,hewasthinkingaboutlogic,notcombinatorics.

Graham’sNumber

R.L.GrahamandB.L.Rothschild,Ramseytheory,StudiesinCombinatorics(ed.G.-C.Rota)MathematicalAssociationofAmerica17(1978)80–99.

TheAffairoftheAbove-AverageDriver

In1981O.Svensonsurveyed161SwedishandAmericanstudents,askingthem

toratetheirdrivingabilityandsafetyrelativetotheothersubjects.Forability,69%ofSwedesconsideredthemselvestobeabovethemedian;forsafety,thefigurewas77%.ThefiguresfortheAmericanstudentswere93%forabilityand88%forsafety.HavingpassedtwoAmericandrivingtests,oneofwhichdidnotinvolvegettinginthecar,Icanseewhytheyoverestimatedtheirabilitiestosuchanextent.SeeO.Svenson,Arewealllessriskyandmoreskillfulthanourfellowdrivers?ActaPsychologica47(1981)143–148.Thiseffectoccursformanyothertraits—popularity,health,memory,job

performance,evenhappinessinrelationships.It’snotespeciallysurprising:it’sonewaypeoplemaintaintheirownself-esteem.Andpoorself-esteemcanbeasignofpsychologicalinadequacy—soinordertobehappyandhealthywehaveevolvedtooverestimatehowhappyandhealthyweare.Dunnoaboutyou,butI’mfeelinggreat.

TheBafflehamBurglary

“Thenumbersare4and13,”saidSoames.“Howutterlyamazing.I—”“Youknowmymethods,Watsup.”“Nevertheless,Ifindittotallyremarkablethatyoucandeducetheanswer

fromsuchavagueconversation.”“Hmm.Weshallsee.Theessence,Watsup,isthateachstatementwemake

addsextrainformationthatwebothknow.Andknowwebothknow,andsoon.Supposetheproductofthetwonumbersispandtheirsumiss.Initiallyyouknowp,whereasIknows.Weeachknowthattheotherknowswhatheknows,butwedonotknowwhatthatis.“Sinceyoudonotknowthetwonumbers,pcannotbeaproductoftwo

primes,suchas35.Forthisis5×7,andthereisnootherwaytoexpressitasaproductofnumbersgreaterthan1,soyouwouldimmediatelydeducethetwonumbers.Forsimilarreasonsitcannotbethecubeofaprime,suchas53=125,sincethisfactorisesonlyas5×25.”“Yes,Iseethat,”Ireplied.“Moresubtly,pcannotbeequaltoqmwhereqisprimeandmiscomposite,

providedthatwheneverddividesmandisgreaterthan1,qdisgreaterthan100.”“Yeeeesss…““Forexample,pcannotbe67×3×5,whichfactorisesinthreeways:67×15,

201×5,and335×3.Sincethelasttwousenumbersgreaterthan100,theycanbeignored,leavingonly67and15asthetwonumbers.”“Ah.Quite.”“Now,yourremarkmakesmeawareofthosefacts,butatthatpointIhave

alreadydeducedthesameinformationfrommysum.Infact,sisnotthesumoftwosuchnumbers.Butyouthenbecomeawareofthatfact,becauseItellyou,soyouthenknowwhattoyouisnewinformationabouts.Ofcourse,webothhavetobearinmindthatifs=200thenthenumbersmustbothbe100,andifs=199theyare100and99.”“Ofcourse.”“Onceyouhaveeliminatedtheimpossible…”saidSoames,“allthatremains

turnsouttobesumssequaltooneofthenumbers11,17,23,27,29,35,37,41,47,51,or53.”“Butpreviouslyyouhavemadescathingremarksabout—”“Oh,itworkswellenoughinmathematics,”hesaidairily.“Fortherewecan

beconfidentthattheimpossiblereallyisimpossible.“Now,attherelevantstageinthededuction,webothknowwhatIhavejust

toldyou.Atwhichjunctureyoupromptlyannouncethatyoucandeducethenumbers!SoIquicklyrunthroughallpossiblepairsofnumberswiththosesums,andIfindthattenoftheelevenpossibilitiesforsshareapossibleproductwithadifferentvalueofs.Sinceyouhavetoldmeyounowknowthenumbers,alltencanbeeliminatedfromourinvestigation.Whichleavesonlyonepossiblesum,17,andonlyoneproductnotoccurringfortwodifferentvaluesofs.Namely,52,whichariseswhenwesplit17as4+13,andonlyinthatmanner.Thereforethetwonumbersmustbe4and13.”Icongratulatedhimonhisperspicacity.“SendaBakerStreetIrreducibletoRouladewiththismessage,”heinstructed,

scribblingthenumbersonascrapofpaper.“Hewillhavetwoarrestswithinthehour.”

Malfatti’sMistake

In1930HymanLobandHerbertRichmondprovedthatthegreedyalgorithmisbetterthanMalfatti’sarrangementinsomecases.HowardEvesnoticedin1946thatforanisoscelestrianglewithaverysharpapex,thestackedsolutionhasalmosttwicetheareaoftheMalfattiarrangement.In1967M.GoldbergprovedthatthegreedyalgorithmalwaysdoesbetterthanMalfatti’sarrangement.In1994VictorZalgallerandG.A.Losprovedthatitalwaysproducesthelargestarea.

HowtoStopUnwantedEchoes

M.R.Schroeder,Diffusesoundreflectionbymaximum-lengthsequence.JournaloftheAcousticalSocietyofAmerica57(1975)149–150.

TheEnigmaoftheVersatileTile

Tenshapesformedfromversa-tiles

TheThrackleConjecture

JánosPachandEthanSterling,Conway’sconjectureformonotonethrackles,AmericanMathematicalMonthly118(June/July2011)544–548.

ATilingThatIsNotPeriodic

Howtotileperiodicallyusingthe7-gon

TheTwoColourTheorem

Havingrackedmybrainsforthreehours,IbeggedSoamestorevealhissecret.“Butthenyouwilltellmehowabsurdlysimpleitis.”“Nay!Never!”“Ibegtodiffer,Watsup.Becauseforonceitisabsurdlysimple.”Silence

stretcheduntilherelented.“Verywell.Assumethattheonlyavailablecoloursareblackandgrey,withwhiteforregionsasyetunconsidered.Letusbeginbycolouringoneregionblack(seetopleftfigureonpage294).ThenIchooseoneadjacentregion,andcolouritgrey(topmiddle).ThenIcolouranadjacentregionblack,thenanothergrey,andsoon.”“Iseethatafterthefirstchoice,successivechoicesareforced,”Isaid

hesitantly.“Yes!Thesolution,ifitexists,mustbeunique—saveforinterchangingthe

twocolours.Andyouseethateventuallythewholemapiscoloured,usingonlyblackandgrey.Sointhiscase,atleast,asolutiondoesexist.”“Agreed.ButIdon’tentirelysee—”“Why.Anexcellentremark.Foronce,mydearWatsup,youhavehitthenail

firmlyonitshead,notyourownthumb.Theproblemistoprovethatanysuchchainofcolouringsleadstothesameoutcome,yes?Becausethatway,theprocesscanneverterminatewithasituationforwhichthenextremainingregioncannotbecoloured.”

Thefirstfewstagesincolouringthemap

“IthinkIseethat.”“Itcanbedone,”saidSoames.“Butthereisasimplermethod.Observethat

everytimewecrossacommonboundary,thecolourchanges.Thereforeifwecrossanoddnumberofboundaries,wemustchoosegrey,andifwecrossanevennumberofboundaries,wemustchooseblack.”Inodded.“But…howcanwebesurethatthereisnoinconsistency?”I

blurted.Soamesgaveabriefgrin.“BecausewecantakeahintfromwhatIhavejust

said,andprescribetheexactcolourofeveryregion.Merelycounthowmanycirclescontainagivenpoint—notonanycircumference,ofcourse,becausewedonotcolourthose.Ifthatnumberiseven,colourthepointblack;ifodd,colouritgrey.“Now,crossinganyboundarylineeitheraddsonecontainingcircle,or

subtractsone.Eitherway,oddchangestoevenandeventoodd,sothetwocoloursoneithersideofthatboundaryaredifferent.”

Numberingregionsaccordingtohowmanycirclestheylieinside.Notehowparity(odd/even)changesacrossboundaries.

Theproofwasasclearasdaylight.“Why,Soames—”“Ofcourse,”heinterrupted,withthebaresthintofasmile,“someofthe

circlesmaybetangenttoothers.Butthesamemethodstillapplies,suitablyinterpreted.Onemustavoidcrossingaboundarylineatapointoftangency,andalittlethoughtshowsthatthiscanalwaysbedone.”Well,maybenotquitedaylight,but…yes,Iunderstood.“Itis—”Ibegan;

thenpaused,seeinghisexpression.“Veryclever,”Ifinished.

TheFourColourTheoreminSpace

Fourequalspherescanbearrangedsothateachtouchestheotherthree.Placethreetomakeanequilateraltriangle,touchingeachother,andthenputthefourthontoptofitintothecentraldimple,makingatetrahedron.Asmallersphere,ofexactlytherightsize,cannowbeplacedinthemiddletotouchallfour.Sowehavefivespheres,eachtouchingtheotherfour,andthesemustallhavedifferentcolours.

Fittinginthefifthsphere

TheGreekIntegrator

Answerfirst.Wehavetosolvetheequation .Divideby4πr2toget.Thereforer=3.Nowforthepalimpsest.

Left:AtypicalpageoftheArchimedespalimpsest.Thethirteenth-centuryreligioustextrunsvertically;thefainteroriginalrunshorizontally.Right:Cleaned-uppagewithclearmathematicaldiagrams.

Archimedes’originalmanuscripthasnotsurvived,butthiscopy(nodoubttheculminationofaseriesofcopies)wasmadebyaByzantinemonkaroundAD950.In1229itwasunboundandscrubbed(fairly)clean,alongwithatleastsixothermanuscripts.Thesheetswerefoldedinhalfandusedtowritea177-pageChristianliturgicaltext—adescriptionoftheproceduresforreligiousservices.Inthe1840sConstantinvonTischendorf,aGermanBiblicalscholar,came

acrossthistextinConstantinople(nowIstanbul),noticedfaintmathematicalwritings,andbroughtapageofithome.In1906aDanishscholar,JohanHeiberg,realisedthatpartofthepalimpsestwasaworkofArchimedes.Hephotographedit,publishingsomeextractsin1910and1915.ThomasHeathtranslatedthematerialshortlyafterwards,butitattractedlittleattention.Inthe1920sthedocumentwasinthepossessionofaFrenchcollector;by1998ithadsomehowmadeitswaytotheUSA,becomingthesubjectofacourtcasebetweenChristie’sauctionhouseandtheGreekOrthodoxChurch,whichallegedthatthedocumenthadbeenstolenfromamonasteryin1920.ThejudgeruledinChristie’sfavouronthegroundsthatthedelaybetweentheallegedtheftandthelegalactionhadbeentoolong.Thedocumentwaspurchasedbyananonymousbuyer(reportedbyDerSpiegeltobeAmazon’sfounderJeffBezos)for$2million.Between1999and2008thedocumentwasconservedattheWaltersArtMuseum,Baltimore,andanalysedbyateamofimagingscientiststoenhancethehiddenwriting.Archimedes’methodcanbeexplained(usingmodernlanguageand

symbolism)asfollows.Startwithasphereofradius1,itscircumscribedcylinder,andacone.Ifweplacethecentreofthesphereatpositionx=1ontherealline,thenthecross-sectionalradiusatanyxbetween0and2is ,anditsmassisproportionaltoπtimesthesquareofthis,namelyπx(2–x)=2πx–πx2.Next,consideraconeobtainedbyrotatingtheliney=xaroundthex-axis,

againfor0≤x≤2.Thecrosssectionatxisacircleofradiusx,andhasareaπx2.Itsmassisproportionaltothis,withthesameconstantofproportionality,sothecombinedmassofthesliceofsphereandthesliceoftheconeis(2πx–πx2)=πx2=2πx.Placethetwoslicesatx=–1,distance1totheleftoftheorigin.Bythelaw

ofthelever,theyareexactlybalancedbyacircleofradius1placeddistancextotheright.Nowmoveallslicesofthesphereandtheconetothesamepointx=–1,so

thattheirtotalmassisconcentratedatthissinglepoint.Thecorresponding(andbalancing)circlesallhaveradius1,andareplacedatalldistancesfrom0to2.Theythereforeformacylinder.Itscentreofmassisinthemiddle,atx=1.

Therefore,bythelawofthelever,

WhatArchimedesdid.Top:Sliceasphere,cone,andcylinder(shownincrosssection:sphere=circle,cone=triangle,cylinder=square)likeloavesofslicedbread.Thenthevolumeofasliceofthecylinder(grey)isthesumofthevolumesofthecorrespondingslicesofthesphereandthecone.Theslicesherehavenonzerothickness,whichintroduceserrors.Archimedesthoughtaboutinfinitelythinslices,forwhichtheerrorsbecomeassmallasweplease.Bottom:theweighingargumentthatrelatesthethreevolumes.Slicesatxofsphereandcone,placedat–1,balancesliceofcylinder

placedatx.

massofsphere+massofcone=massofcylinder

andsincemassisproportionaltovolume,

volumeofsphere+volumeofcone=volumeofcylinder.

However,Archimedesalreadyknewthatthevolumeoftheconeisonethirdthatofthecylinder(onethirdareaofbasetimesheight,remember?),sothevolumeofthesphereistwothirdsthatofthecylinder.Thevolumeofthecylinderisthe

areaofthebase(πr2)timestheheight(2r),thatis,2πr3.Sothevolumeofthesphereis ofthis,namely .Archimedesderivedthesurfaceareaofthespherebyasimilarprocedure.Hedescribedtheprocessgeometrically,butit’seasiertofollowtheargument

usingmodernnotation.Consideringthathedidallthisaround250BC,andthathealsodevelopedthelawofthelever,it’sanamazingachievement.

WhytheLeopardGotItsSpots

W.L.Allen,I.C.Cuthill,N.E.Scott-Samuel,andR.J.Baddeley.Whytheleopardgotitsspots:relatingpatterndevelopmenttoecologyinfelids,ProceedingsoftheRoyalSocietyB:BiologicalSciences278(2011)1373–1380.

PolygonsForever

Itlooksasthoughthefigurewillgrowwithoutlimit,butactuallyitremainswithinaboundedregionoftheplane:acirclewhoseradiusisabout8.7.Theratiooftheradiiofacirclecircumscribedaboutaregularn-gonandone

inscribedinitissecπ/n,wheresecisthetrigonometricsecantfunctionandI’musingradianmeasurefortheangle.(Replaceπby180°fordegreemeasure.)Soforeachntheradiusofthecirclecircumscribedabouttheregularn-goninthepictureis

S=secπ/3×secπ/4×secπ/5×…×secπ/n

Wewantthelimitofthisproductasntendstoinfinity.Takelogarithms:

logS=logsecπ/3+logsecπ/4+logsecπ/5+…+logsecπ/n

Whenxissmall,logsecx~x2/2,sothisseriescanbecomparedwith

1/32+1/42+1/52+…+1/n2

whichconvergesasntendstoinfinity.ThereforelogSisfinite,soSisfinite.Thesumofthetermsupton=1,000,000yields8.7asareasonableestimate.Ilearnedaboutthisproblem,andtheanswergivenabove,fromabookreview

byHaroldBoas[AmericanMathematicalMonthly121(2014)178–182].HetraceditbackasfarasMathematicsandtheImaginationbyEdwardKasnerand

JamesNewmanin1940.Hewrites:“Perhapsthisamusingexamplewillbecomepartofthestandardloreifthefiguregetsreproducedinenoughbooks.”I’mtrying,Harold.

TheAdventureoftheRowingMen

SoamesandIfoundtwofurtherarrangements,notcountingreflectionsinthecentrelineasdifferent:

Arrangements0167and0356

“Forallthemechanicalcomplexityoftheproblem,”saidSoames,“itreducesultimatelytomerearithmetic.Wehavetosplitthenumbersfrom0to7intotwosets,eachwithsum14.”“Ifweknowonesuchset,theotherisdeterminedandalsohassum14.”“Yes,Watsup,thatisevident:listthenumbersthatarenotinthefirstset.”“Iagreeitistrivial,Soames,butitimpliesthatwecanworkontheset

containing0,whichamountstoputtingthesternoarontheleft,whichwemayassumebyapplyingareflectionifnecessary.Therebyreducingthenumberofcasestobeconsidered.”“True.”Nowthedeductionsalmostmadethemselves.“Ifthesetalsocontains1,”I

pointedout,“thentheothertwoaddto13,sotheymustbe6and7,giving0167.Ifitdoesnotcontain1butdoescontain2,thentheonlypossibilityis0257.Ifitstarts03therearetwopossibilities:0347and0356.Wecandismissarrangementsthatstart04becauseitisnotpossibletomake10fromtwoofthenumbers5,6,7.Asimilarargumentdisposesof05,06,and07.”“Soyouhavededuced,”Soamessummarised,“thattheonlypossibilities,

excludingleft–rightreflections,are

0167025703560347

Now0257istheGermanrigand0347theItalian.Therearetwoothers:thosethatIhavelaidoutwithmymatchst—”Hesuddenlysatup,rigid.“GreatScott!”“What,Soames?”

“Ithasjuststruckme,nopunintended,Watsup,thatthismatch—”hewaveditatme—“isnotarareearlyCongreve,asIhadimagined,butoneofIrinyi’snoiselessmatches.Whenhischemistryprofessorblewhimselfup,Irinyiwasinspiredtoreplacepotassiumchloratebyleaddioxideintheheadofthematch.”“Ah.Isthatsignificant,Soames?”“Mostassuredly,Watsup.Itcastsanentirelynewlight—againnopun

intended—ononeofourmostbizarreunsolvedcases.”“TheRemarkableAffairoftheUpsideDownTeapot!”Icried.“Youhaveit,

Watsup.Now,ifyournotesrecordwhetherthematchwasdroppedtotheleftortherightofthemummifiedparrot…”Soames’sanalysisisbasedon:MauriceBrearley,‘Oararrangementsinrowingeights’,inOptimalStrategies

inSports(ed.S.P.LadanyandR.E.Machol),North-Holland1977.JohnBarrow,OneHundredEssentialThingsYouDidn’tKnowYouDidn’t

Know,W.W.Norton,NewYork2009.AsSoameswarned,itisaninitialsimplifiedapproachtoahighlycomplex

issue.Bytheway,the1877BoatRacewasadeadheat—theonlyoneinthehistory

oftheevent.

RingsofRegularSolids

JohnMasonandTheodorusDekkerfoundsimplermethodsthanŚwierczkowski’stoprovetheimpossibility.Wheneveryougluetwoidenticaltetrahedronsbytheirfaces,eachisareflectionoftheotherinthecommonface.

Twotetrahedronswithacommonface(shaded).Eachisareflectionoftheotherinthisface.

Startwithonetetrahedron.Ithasfourfaces,sotherearefoursuchreflections;

callthemr1,r2,r3,andr4.Eachreflectionsendseverythingbackwhereitstartedifyoudoittwice,sor1r1=e,whereeisthetransformation‘donothing’.Thesamegoesfortheotherreflections.Soallcombinationsofseveralreflectionsareproductslike

r1r4r3r4r2r1r3r1

wherethesequenceofsubscripts14342131canbeanysequenceformedbythefournumbers1,2,3,4inwhichnonumbereveroccurstwiceinarow.Forexample,14332131isnotpermitted.Thereasonisthatherer3r3isthesamereflectionperformedtwice,soitisthesamease,whichhasnoeffectandcanthereforebedeleted.Ifachainclosesup,onefurtherreflectionappliedtothelasttetrahedroninthe

chainproducesatetrahedronthatcoincideswiththeinitialone.Sowegetanequationlike

r1r4r3r4r2r1r3r1=e

(onlylongerandmorecomplicated)whereestandsfor‘donothing’.Bywritingdownformulasforthefourreflections,andusingsuitablealgebraicmethods,itcanbeprovedthatnosuchequationholds.Fordetails,see:T.J.Dekker,OnreflectionsinEuclideanspacesgeneratingfreeproducts,

NieuwArchiefvoorWiskunde7(1959)57–60.M.ElgersmaandS.Wagon,ClosingaPlatonicgap,Mathematical

Intelligencerinthepress.J.H.Mason,Canregulartetrahedronsbegluedtogetherfacetofacetoforma

ring?MathematicalGazette56(1972)194–197.H.Steinhaus,Problem175,ColloquiumMathematicum4(1957)243.S.Świerczkowski,OnafreegroupofrotationsoftheEuclideanspace,

IndagationesMathematicae20(1958)376–378.S.Świerczkowski,Onchainsofregulartetrahedra,Colloquium

Mathematicum7(1959)9–10.

TheImpossibleRoute

“Asyousorightlysay,youdonotseeit,”saidSoames.“Youknowmymethods:usethem.”“Verywell,Soames,”Ireplied.“Youhavealwaysinstructedmetodiscard

thatwhichisirrelevant.Ishallthereforerepeatmyanalysis,andtoeliminateanyconceivablepossibilityoferrorIshallrepresenttheprobleminitssimplestform.Inumbertheregionsonthemap—likeso.Therearefiveofthem.ThenIdrawadiagram—Ibelieveitiscalledagraph—showingtheregionsandtheirconnectionsinschematicform.”Heremainedsilent,hisexpressionunreadable.“Wemustproceedfromregion1toregion5,leavingbridgeAtothelast.

Startingfrom1,theonlyalternativeistocrossbridgeB,andthenCandDareforceduponus.WemustuseeitherbridgeEorF.LetussaywetakeE.WecannotuseFbecausethattakesustoregion4andwecannotproceedfurther.However,wecannotthenuseA,becausethattakesustoregion1andwecannotproceedfurther.ThesamegoesifweemployFinplaceofE.Irestmycase.”

Left:Watsup’sfiveregions.Right:Graphofconnections.

“Why,Watsup?”“Because,Soames,Ihaveeliminatedtheimpossible.”Heraisedoneeyebrow.

“Sowhatremains,howeverunlikely,”Icontinued,“mustbe—”“Goon.”“ButSoames,nothingremains!Thereforetheproblemhasnosolution!”“Wrong.Ihavetoldyouthatthereareeight.”“Thenyoumusthaveliedabouttheconditions.”

“Ididnot.”“ThenIamstumped.WhathaveImissedout?”“Nothing.”“But—”“Youputtoomuchin,Watsup.Youmadeanunwarrantedassumption.Your

errorwastoassumethatthepathdoesnotleavethemap.”“ButyoutoldmethattheriverscontinuetoflowtotheSwissborders,andwe

arenotpermittedtorecrosstheborder.”“Yes.ButthemapdoesnotdepictthewholeofSwitzerland.Wheredoesthe

rivercomefrom?”“D’oh!”Istruckmyforeheadwithmyhand.“Dough?”“Merelyaninadvertentexpressionberatingmyselfformyownstupidity,

Soames.Not‘dough’.Morealongthelinesof‘D’oh!’”“Iadviseyoutoavoidit,Watsup.Itdoesnotbecomeyou,anditwillnever

catchon.”“Asyousay,Soames.Whatcausedmyoutburstwastherealisationthatwe

cancompletemysecondattemptbyencirclingthesourceoftheriverandpassingoverbridgeA.”“Correct.”“Soregions1and4inmyfigureareactuallythesameregion.”“Indeed.”

Soames’sroute

“That,”Isaidafteramoment,“wasunfair.HowamItoknowthattheriverriseswithinSwissborders?Thesourcewasnotshownonyourmap.”“Because,Watsup,Itoldyouthatthereisatleastoneroutesatisfyingmy

conditions.ItfollowsthatthesourcemustlieinSwitzerland.”Touché.ThenIrememberedthathehadreferredtoeightroutes.“Iseea

secondroute,Soames:interchangebridgesEandF.ButIconfesstheothersixeludeme.”“Ah.YourassertionthatwemustbeginwithbridgeBisnolongervalidwhen

regions1and4aremerged.Letmeredrawyoursimplifiedfigurecorrectly.”

Soames’scorrectedgraph

“IhavedrawnbridgeAasadottedlineasareminderthatwemustleaveAuntillast.Observe:startingfromregion1,thebridgesotherthanAformtwodistinctloops:BCDandEF.Wecantraverseeachloopintwodirections:BCDorDCB,andEForFE.Furthermore,wecanstartwitheitherloopandthentraversetheother.Finally,wemustappendbridgeA.Sothedifferentroutesare

BCD–EF–ADCB–EF–ABCD–FE–ADCB–FE–AEF–BCD–AEF–DCB–AFE–BCD–AFE–DCB–A

“Atotalofeight.”“Iseemyerrorclearlynow,Soames,”Iadmitted.“Youseeyourspecificerror,Watsup,butnottheunderlyinggenerality,which

afflictsallargumentsabouteliminatingtheimpossible.”Ishookmyhead,puzzled.“Whatdoyoumean?”“Imean,Watsup,thatyoudidnotconsiderallpossibilities.Andthereason

was—”

AgainIstruckmyheadwithmyhand,butthistimeIrefrainedfromutteringasound,notwishingtobethebuttofSoames’sscorn.“Iforgottothinkoutsidethebox.”

TheGreatMathematicalProblemsIANSTEWART

‘Ianreallyisunsurpassedasraconteuroftheworldofnumbers.Heguidesusonamind-bogglingjourneyfromtheultra-trivialtotheprofound.’NewScientist

TheGreatMathematicalProblemsdescribesthereallybigproblemsthatdefinemathematics:whytheyexist,whytheymatter,whatdrivesmathematicianstoincrediblelengthstosolvethemandwheretheystandinthecontextofmathematicsandscienceasawhole.Itcontainssolvedproblems–likethePoincaréConjecture,crackedbytheeccentricgeniusGrigoriPerelman,whorefusedacademichonoursandamillion-dollarprizeforhiswork,andoneswhich,liketheRiemannHypothesis,remainbafflingaftercenturies.

Stewartistheguidetothismysteriousandexcitingworld,showinghowmodernmathematiciansconstantlyrisetothechallengessetbytheirpredecessors,asthegreatmathematicalproblemsofthepastsuccumbtothenewtechniquesandideasofthepresent.

March2013

PaperbackISBN9781846683374

eBookeISBN9781847653512

MathematicsofLife

IANSTEWART

‘Asalways,[Stewart]explainscomplicatedmathematicalideasbrilliantly’NewScientist

Anewpartnershipofbiologistsandmathematiciansisusingmathsasneverbefore:tounlockthesecretsoflifeonearth.Theyareuncoveringthehiddentruthsbehindthebehaviourofanimalsandplants–andtheecologicalbalanceofourentireplanet.CelebratedmathematicianIanStewartprovidesexamplesfromthemysteriousoperationsofthebraintothepuzzlingbehaviourofviruses.Inbetween,heinvestigatesthecreationofartificiallifeandtheprobabilityofaliensandanswersintriguingquestionslikehowthetigergotitsstripesandwhymatinglizardsplayrock-paper-scissors.

Together,mathematiciansandbiologistsareansweringsomeofthemostcrucialquestionsinscience:thisbookisaninsider’sviewofwhatthey’redoing–andwherethatknowledgeistakingus.

July2013Paperback

ISBN9781846682056eBook

eISBN9781847653505

ProfessorStewart’sIncredibleNumbersIANSTEWART

ThenewmathematicalbestsellerfromProfessorIanStewart

IanStewartexplorestheastonishingpropertiesofnumbersfrom1to10tozeroandinfinity,includingonefigurethat,ifyouwroteitout,wouldspantheuniverse.Helooksateverykindofnumberyoucanthinkof–real,imaginary,rational,irrational,positiveandnegative–alongwithseveralyoumighthavethoughtyoucouldn’tthinkof.Heexplainstheinsightsoftheancientmathematicians,showshownumbershaveevolvedthroughtheages,andrevealsthewaynumericaltheoryenableseverydaylife.

UnderProfessorStewart’sguidanceyouwilldiscoverthemathematicsofcodes,sudoko,Rubik’scube,music,primesandpi.Youmaybesurprisedtofindyouliveineleven-dimensionalspace,thatofthetwenty-threepeopleonafootballpitchtwoaremorelikelythannottosharethesamebirthday,andthatforty-twoisaveryinterestingnumber.

ProfessorStewart’sIncredibleNumberswilldelighteveryonewholovesnumbers–includingthosewhocurrentlythinktheydon’t.

Availablenowfromtheappstore

17EquationsthatChangedtheWorldIANSTEWART

Auniquehistoryofhumanitytoldthroughitsseventeendefiningequations;fromPythagorastoCalculus

FromNewton’sLawofGravitytotheBlack-Scholesmodelusedbybankerstopredictthemarkets,equations,areeverywhere–andtheyarefundamentaltoeverydaylife.

SeventeenEquationsthatChangedtheWorldexaminesseventeengroundbreakingequationsthathavealteredthecourseofhumanhistory.HeexploreshowPythagoras’sTheoremledtoGPSandSatNav;howlogarithmsareappliedinarchitecture;whyimaginarynumberswereimportantinthedevelopmentofthedigitalcamera,andwhatisreallygoingonwithSchrödinger’scat.

Entertaining,surprisingandvastlyinformative,SeventeenEquationsthatChangedtheWorldisahighlyoriginalexploration–andexplanation–oflifeonearth.

February2012Paperback

ISBN9781846685323eBook

eISBN9781847657695

ProfessorStewart’sHoardofMathematicalTreasuresIANSTEWART

‘AnidealpresentforanyoneaddictedtoSudoku-likepuzzlesandbeginningtowonderwhatmightliebeyond’Spectator

November2010

BPB352ppISBN9781846683466

ProfessorStewart’sCabinetofMathematicalCuriositiesIANSTEWART

‘StewarthasserveduptheinstructiveequivalentofaMichelin-starredtastingmenu’GuardianJuly2010BPB320ppISBN9781846683459