Archived Problems - Project Euler

5/2/2015 Archived Problems - Project Euler

https://projecteuler.net/show=all 1/163

Problem1:Multiplesof3and5

Ifwelistallthenaturalnumbersbelow10thataremultiplesof3or5,weget3,5,6and9.Thesumofthesemultiplesis23.

Findthesumofallthemultiplesof3or5below1000.

Problem2:EvenFibonaccinumbers

EachnewtermintheFibonaccisequenceisgeneratedbyaddingtheprevioustwoterms.Bystartingwith1and2,thefirst10termswillbe:

1,2,3,5,8,13,21,34,55,89,...

ByconsideringthetermsintheFibonaccisequencewhosevaluesdonotexceedfourmillion,findthesumoftheevenvaluedterms.

Problem3:Largestprimefactor

Theprimefactorsof13195are5,7,13and29.

Whatisthelargestprimefactorofthenumber600851475143?

Problem4:Largestpalindromeproduct

Apalindromicnumberreadsthesamebothways.Thelargestpalindromemadefromtheproductoftwo2digitnumbersis9009=9199.

Findthelargestpalindromemadefromtheproductoftwo3digitnumbers.

Problem5:Smallestmultiple

2520isthesmallestnumberthatcanbedividedbyeachofthenumbersfrom1to10withoutanyremainder.

Whatisthesmallestpositivenumberthatisevenlydivisiblebyallofthenumbersfrom1to20?

Problem6:Sumsquaredifference

Thesumofthesquaresofthefirsttennaturalnumbersis,

12+22+...+102=385

Thesquareofthesumofthefirsttennaturalnumbersis,

(1+2+...+10)2=552=3025

Hencethedifferencebetweenthesumofthesquaresofthefirsttennaturalnumbersandthesquareofthesumis3025385=2640.

Findthedifferencebetweenthesumofthesquaresofthefirstonehundrednaturalnumbersandthesquareofthesum.

Problem7:10001stprime

Bylistingthefirstsixprimenumbers:2,3,5,7,11,and13,wecanseethatthe6thprimeis13.

Whatisthe10001stprimenumber?



Problem8:Largestproductinaseries

Thefouradjacentdigitsinthe1000digitnumberthathavethegreatestproductare9989=5832.

7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450

Findthethirteenadjacentdigitsinthe1000digitnumberthathavethegreatestproduct.Whatisthevalueofthisproduct?

Problem9:SpecialPythagoreantriplet

APythagoreantripletisasetofthreenaturalnumbers,a



Whatisthegreatestproductoffouradjacentnumbersinthesamedirection(up,down,left,right,ordiagonally)inthe2020grid?

Problem12:Highlydivisibletriangularnumber

Thesequenceoftrianglenumbersisgeneratedbyaddingthenaturalnumbers.Sothe7thtrianglenumberwouldbe1+2+3+4+5+6+7=28.Thefirsttentermswouldbe:

1,3,6,10,15,21,28,36,45,55,...

Letuslistthefactorsofthefirstseventrianglenumbers:

1:13:1,36:1,2,3,610:1,2,5,1015:1,3,5,1521:1,3,7,2128:1,2,4,7,14,28

Wecanseethat28isthefirsttrianglenumbertohaveoverfivedivisors.

Whatisthevalueofthefirsttrianglenumbertohaveoverfivehundreddivisors?

Problem13:Largesum

Workoutthefirsttendigitsofthesumofthefollowingonehundred50digitnumbers.

37107287533902102798797998220837590246510135740250463769376774900097126481248969700780504170182605387432498619952474105947423330951305812372661730962991942213363574161572522430563301811072406154908250230675882075393461711719803104210475137780632466768926167069662363382013637841838368417873436172675728112879812849979408065481931592621691275889832738442742289174325203219235894228767964876702721893184745144573600130643909116721685684458871160315327670386486105843025439939619828917593665686757934951621764571418565606295021572231965867550793241933316490635246274190492910143244581382266334794475817892575867718337217661963751590579239728245598838407582035653253593990084026335689488301894586282278288018119938482628201427819413994056758715117009439035398664372827112653829987240784473053190104293586865155060062958648615320752733719591914205172558297169388870771546649911559348760353292171497005693854370070576826684624621495650076471787294438377604532826541087568284431911906346940378552177792951453612327252500029607107508256381565671088525835072145876576172410976447339110607218265236877223636045174237069058518606604482076212098132878607339694128114266041808683061932846081119106155694051268969251934325451728388641918047049293215058642563049483624672216484350762017279180399446930047329563406911573244438690812579451408905770622942919710792820955037687525678773091862540744969844508330393682126183363848253301546861961243487676812975343759465158038628759287849020152168555482871720121925776695478182833757993103614740356856449095527097864797581167263201004368978425535399209318374414978068609844840309812907779179908821879532736447567559084803087086987551392711854517078544161852424320693150332599594068957565367821070749269665376763262354472106979395067965269474259770973916669376304263398708541052684708299085211399427365734116182760315001271653786073615010808570091499395125570281987460043753582903531743471732693212357815498262974255273730794953759765105305946966067683156574377167401875275889028025717332296191766687138199318110487701902712526768027607800301367868099252546340106163286652636270218540497705585629946580636237993140746255962240744869082311749777923654662572469233228109171419143028819710328859780666976089293863828502533340334413065578016127815921815005561868836468420090470230530811728164304876237919698424872550366387845831148769693215490281042402013833512446218144177347063783299490636259666498587618221225225512486764533677201869716985443124195724099139590089523100588229554825530026352078153229679624948164195386821877476085327132285723110424803456124867697064507995236377742425354112916842768655389262050249103265729672370191327572567528565324825826546309220705859652229798860272258331913126375147341994889534765745501



184957014548792889848568277260777137214037988797153829820378303147352772158034814451349137322665138134829543829199918180278916522431027392251122869539409579530664052326325380441000596549391598795936352974615218550237130764225512118369380358038858490341698116222072977186158236678424689157993532961922624679571944012690438771072750481023908955235974572318970677254791506150550495392297953090112996751986188088225875314529584099251203829009407770775672113067397083047244838165338735023408456470580773088295917476714036319800818712901187549131054712658197623331044818386269515456334926366572897563400500428462801835170705278318394258821455212272512503275512160354698120058176216521282765275169129689778932238195734329339946437501907836945765883352399886755061649651847751807381688378610915273579297013376217784275219262340194239963916804498399317331273132924185707147349566916674687634660915035914677504995186714302352196288948901024233251169136196266227326746080059154747183079839286853520694694454072476841822524674417161514036427982273348055556214818971426179103425986472045168939894221798260880768528778364618279934631376775430780936333301898264209010848802521674670883215120185883543223812876952786713296124747824645386369930090493103636197638780396218407357239979422340623539380833965132740801111666627891981488087797941876876144230030984490851411606618262936828367647447792391803351109890697907148578694408955299065364044742557608365997664579509666024396409905389607120198219976047599490197230297649139826800329731560371200413779037855660850892521673093931987275027546890690370753941304265231501194809377245048795150954100921645863754710598436791786391670211874924319957006419179697775990283006991536871371193661495281130587638027841075444973307840789923115535562561142322423255033685442488917353448899115014406480203690680639606723221932041495354150312888033953605329934036800697771065056663195481234880673210146739058568557934581403627822703280826165707739483275922328459417065250945123252306082291880205877731971983945018088807242966198081119777158542502016545090413245809786882778948721859617721078384350691861554356628840622574736922845095162084960398013400172393067166682355524525280460972253503534226472524250874054075591789781264330331690

Problem14:LongestCollatzsequence

Thefollowingiterativesequenceisdefinedforthesetofpositiveintegers:

nn/2(niseven)n3n+1(nisodd)

Usingtheruleaboveandstartingwith13,wegeneratethefollowingsequence:

134020105168421

Itcanbeseenthatthissequence(startingat13andfinishingat1)contains10terms.Althoughithasnotbeenprovedyet(CollatzProblem),itisthoughtthatallstartingnumbersfinishat1.

Whichstartingnumber,underonemillion,producesthelongestchain?

NOTE:Oncethechainstartsthetermsareallowedtogoaboveonemillion.

Problem15:Latticepaths

Startinginthetopleftcornerofa22grid,andonlybeingabletomovetotherightanddown,thereareexactly6routestothebottomrightcorner.

Howmanysuchroutesaretherethrougha2020grid?



Problem16:Powerdigitsum

215=32768andthesumofitsdigitsis3+2+7+6+8=26.

Whatisthesumofthedigitsofthenumber21000?

Problem17:Numberlettercounts

Ifthenumbers1to5arewrittenoutinwords:one,two,three,four,five,thenthereare3+3+5+4+4=19lettersusedintotal.

Ifallthenumbersfrom1to1000(onethousand)inclusivewerewrittenoutinwords,howmanyletterswouldbeused?

NOTE:Donotcountspacesorhyphens.Forexample,342(threehundredandfortytwo)contains23lettersand115(onehundredandfifteen)contains20letters.Theuseof"and"whenwritingoutnumbersisincompliancewithBritishusage.

Problem18:MaximumpathsumI

Bystartingatthetopofthetrianglebelowandmovingtoadjacentnumbersontherowbelow,themaximumtotalfromtoptobottomis23.

374

2468593

Thatis,3+7+4+9=23.

Findthemaximumtotalfromtoptobottomofthetrianglebelow:

759564

17478218358710

2004824765190123750334

880277730763679965042806167092

41412656834080703341487233473237169429

5371446525439152975114701133287773177839681757

917152381714914358502729486366046889536730731669874031

046298272309709873933853600423

NOTE:Asthereareonly16384routes,itispossibletosolvethisproblembytryingeveryroute.However,Problem67,isthesamechallengewithatrianglecontainingonehundredrowsitcannotbesolvedbybruteforce,andrequiresaclevermethod!o)

Problem19:CountingSundays

Youaregiventhefollowinginformation,butyoumayprefertodosomeresearchforyourself.

1Jan1900wasaMonday.ThirtydayshasSeptember,April,JuneandNovember.Alltheresthavethirtyone,SavingFebruaryalone,Whichhastwentyeight,rainorshine.Andonleapyears,twentynine.Aleapyearoccursonanyyearevenlydivisibleby4,butnotonacenturyunlessitisdivisibleby400.

HowmanySundaysfellonthefirstofthemonthduringthetwentiethcentury(1Jan1901to31Dec2000)?

Problem20:Factorialdigitsum



n!meansn(n1)...321

Forexample,10!=109...321=3628800,andthesumofthedigitsinthenumber10!is3+6+2+8+8+0+0=27.

Findthesumofthedigitsinthenumber100!

Problem21:Amicablenumbers

Letd(n)bedefinedasthesumofproperdivisorsofn(numberslessthannwhichdivideevenlyinton).Ifd(a)=bandd(b)=a,whereab,thenaandbareanamicablepairandeachofaandbarecalledamicablenumbers.

Forexample,theproperdivisorsof220are1,2,4,5,10,11,20,22,44,55and110therefored(220)=284.Theproperdivisorsof284are1,2,4,71and142sod(284)=220.

Evaluatethesumofalltheamicablenumbersunder10000.

Problem22:Namesscores

Usingnames.txt(rightclickand'SaveLink/TargetAs...'),a46Ktextfilecontainingoverfivethousandfirstnames,beginbysortingitintoalphabeticalorder.Thenworkingoutthealphabeticalvalueforeachname,multiplythisvaluebyitsalphabeticalpositioninthelisttoobtainanamescore.

Forexample,whenthelistissortedintoalphabeticalorder,COLIN,whichisworth3+15+12+9+14=53,isthe938thnameinthelist.So,COLINwouldobtainascoreof93853=49714.

Whatisthetotalofallthenamescoresinthefile?

Problem23:Nonabundantsums

Aperfectnumberisanumberforwhichthesumofitsproperdivisorsisexactlyequaltothenumber.Forexample,thesumoftheproperdivisorsof28wouldbe1+2+4+7+14=28,whichmeansthat28isaperfectnumber.

Anumberniscalleddeficientifthesumofitsproperdivisorsislessthannanditiscalledabundantifthissumexceedsn.

As12isthesmallestabundantnumber,1+2+3+4+6=16,thesmallestnumberthatcanbewrittenasthesumoftwoabundantnumbersis24.Bymathematicalanalysis,itcanbeshownthatallintegersgreaterthan28123canbewrittenasthesumoftwoabundantnumbers.However,thisupperlimitcannotbereducedanyfurtherbyanalysiseventhoughitisknownthatthegreatestnumberthatcannotbeexpressedasthesumoftwoabundantnumbersislessthanthislimit.

Findthesumofallthepositiveintegerswhichcannotbewrittenasthesumoftwoabundantnumbers.

Problem24:Lexicographicpermutations

Apermutationisanorderedarrangementofobjects.Forexample,3124isonepossiblepermutationofthedigits1,2,3and4.Ifallofthepermutationsarelistednumericallyoralphabetically,wecallitlexicographicorder.Thelexicographicpermutationsof0,1and2are:

012021102120201210

Whatisthemillionthlexicographicpermutationofthedigits0,1,2,3,4,5,6,7,8and9?

Problem25:1000digitFibonaccinumber

TheFibonaccisequenceisdefinedbytherecurrencerelation:

Fn=Fn1+Fn2,whereF1=1andF2=1.

Hencethefirst12termswillbe:

F1=1



F2=1F3=2F4=3F5=5F6=8F7=13F8=21F9=34F10=55F11=89F12=144

The12thterm,F12,isthefirsttermtocontainthreedigits.

WhatistheindexofthefirsttermintheFibonaccisequencetocontain1000digits?

Problem26:Reciprocalcycles

Aunitfractioncontains1inthenumerator.Thedecimalrepresentationoftheunitfractionswithdenominators2to10aregiven:

1/2 = 0.51/3 = 0.(3)1/4 = 0.251/5 = 0.21/6 = 0.1(6)1/7 = 0.(142857)1/8 = 0.1251/9 = 0.(1)1/10 = 0.1

Where0.1(6)means0.166666...,andhasa1digitrecurringcycle.Itcanbeseenthat1/7hasa6digitrecurringcycle.

Findthevalueofd



21222324252078910196121118543121716151413

Itcanbeverifiedthatthesumofthenumbersonthediagonalsis101.

Whatisthesumofthenumbersonthediagonalsina1001by1001spiralformedinthesameway?

Problem29:Distinctpowers

Considerallintegercombinationsofabfor2a5and2b5:

22=4,23=8,24=16,25=3232=9,33=27,34=81,35=24342=16,43=64,44=256,45=102452=25,53=125,54=625,55=3125

Iftheyarethenplacedinnumericalorder,withanyrepeatsremoved,wegetthefollowingsequenceof15distinctterms:

4,8,9,16,25,27,32,64,81,125,243,256,625,1024,3125

Howmanydistincttermsareinthesequencegeneratedbyabfor2a100and2b100?

Problem30:Digitfifthpowers

Surprisinglythereareonlythreenumbersthatcanbewrittenasthesumoffourthpowersoftheirdigits:

1634=14+64+34+44

8208=84+24+04+84

9474=94+44+74+44

Thesumofthesenumbersis1634+8208+9474=19316.

Findthesumofallthenumbersthatcanbewrittenasthesumoffifthpowersoftheirdigits.

Problem31:Coinsums

InEnglandthecurrencyismadeupofpound,,andpence,p,andthereareeightcoinsingeneralcirculation:

1p,2p,5p,10p,20p,50p,1(100p)and2(200p).

Itispossibletomake2inthefollowingway:

11+150p+220p+15p+12p+31p

Howmanydifferentwayscan2bemadeusinganynumberofcoins?

Problem32:Pandigitalproducts

Weshallsaythatanndigitnumberispandigitalifitmakesuseofallthedigits1tonexactlyonceforexample,the5digitnumber,15234,is1through5pandigital.

Theproduct7254isunusual,astheidentity,39186=7254,containingmultiplicand,multiplier,andproductis1through9pandigital.

Findthesumofallproductswhosemultiplicand/multiplier/productidentitycanbewrittenasa1through9pandigital.

HINT:Someproductscanbeobtainedinmorethanonewaysobesuretoonlyincludeitonceinyoursum.

As1=14isnotasumitisnotincluded.



Problem33:Digitcancellingfractions

Thefraction49/98isacuriousfraction,asaninexperiencedmathematicianinattemptingtosimplifyitmayincorrectlybelievethat49/98=4/8,whichiscorrect,isobtainedbycancellingthe9s.

Weshallconsiderfractionslike,30/50=3/5,tobetrivialexamples.

Thereareexactlyfournontrivialexamplesofthistypeoffraction,lessthanoneinvalue,andcontainingtwodigitsinthenumeratoranddenominator.

Iftheproductofthesefourfractionsisgiveninitslowestcommonterms,findthevalueofthedenominator.

Problem34:Digitfactorials

145isacuriousnumber,as1!+4!+5!=1+24+120=145.

Findthesumofallnumberswhichareequaltothesumofthefactorialoftheirdigits.

Problem35:Circularprimes

Thenumber,197,iscalledacircularprimebecauseallrotationsofthedigits:197,971,and719,arethemselvesprime.

Therearethirteensuchprimesbelow100:2,3,5,7,11,13,17,31,37,71,73,79,and97.

Howmanycircularprimesaretherebelowonemillion?

Problem36:Doublebasepalindromes

Thedecimalnumber,585=10010010012(binary),ispalindromicinbothbases.

Findthesumofallnumbers,lessthanonemillion,whicharepalindromicinbase10andbase2.

Problem37:Truncatableprimes

Thenumber3797hasaninterestingproperty.Beingprimeitself,itispossibletocontinuouslyremovedigitsfromlefttoright,andremainprimeateachstage:3797,797,97,and7.Similarlywecanworkfromrighttoleft:3797,379,37,and3.

Findthesumoftheonlyelevenprimesthatarebothtruncatablefromlefttorightandrighttoleft.

Problem38:Pandigitalmultiples

Takethenumber192andmultiplyitbyeachof1,2,and3:

1921=1921922=3841923=576

Byconcatenatingeachproductwegetthe1to9pandigital,192384576.Wewillcall192384576theconcatenatedproductof192and(1,2,3)

Thesamecanbeachievedbystartingwith9andmultiplyingby1,2,3,4,and5,givingthepandigital,918273645,whichistheconcatenatedproductof9and(1,2,3,4,5).

Whatisthelargest1to9pandigital9digitnumberthatcanbeformedastheconcatenatedproductofanintegerwith(1,2,...,n)wheren>1?

Note:as1!=1and2!=2arenotsumstheyarenotincluded.

(Pleasenotethatthepalindromicnumber,ineitherbase,maynotincludeleadingzeros.)

NOTE:2,3,5,and7arenotconsideredtobetruncatableprimes.



Problem39:Integerrighttriangles

Ifpistheperimeterofarightangletrianglewithintegrallengthsides,{a,b,c},thereareexactlythreesolutionsforp=120.

{20,48,52},{24,45,51},{30,40,50}

Forwhichvalueofp1000,isthenumberofsolutionsmaximised?

Problem40:Champernowne'sconstant

Anirrationaldecimalfractioniscreatedbyconcatenatingthepositiveintegers:

0.123456789101112131415161718192021...

Itcanbeseenthatthe12thdigitofthefractionalpartis1.

Ifdnrepresentsthenthdigitofthefractionalpart,findthevalueofthefollowingexpression.

d1d10d100d1000d10000d100000d1000000

Problem41:Pandigitalprime

Weshallsaythatanndigitnumberispandigitalifitmakesuseofallthedigits1tonexactlyonce.Forexample,2143isa4digitpandigitalandisalsoprime.

Whatisthelargestndigitpandigitalprimethatexists?

Problem42:Codedtrianglenumbers

Thenthtermofthesequenceoftrianglenumbersisgivenby,tn=n(n+1)sothefirsttentrianglenumbersare:

1,3,6,10,15,21,28,36,45,55,...

Byconvertingeachletterinawordtoanumbercorrespondingtoitsalphabeticalpositionandaddingthesevaluesweformawordvalue.Forexample,thewordvalueforSKYis19+11+25=55=t10.Ifthewordvalueisatrianglenumberthenweshallcallthewordatriangleword.

Usingwords.txt(rightclickand'SaveLink/TargetAs...'),a16KtextfilecontainingnearlytwothousandcommonEnglishwords,howmanyaretrianglewords?

Problem43:Substringdivisibility

Thenumber,1406357289,isa0to9pandigitalnumberbecauseitismadeupofeachofthedigits0to9insomeorder,butitalsohasaratherinterestingsubstringdivisibilityproperty.

Letd1bethe1stdigit,d2bethe2nddigit,andsoon.Inthisway,wenotethefollowing:

d2d3d4=406isdivisibleby2d3d4d5=063isdivisibleby3d4d5d6=635isdivisibleby5d5d6d7=357isdivisibleby7d6d7d8=572isdivisibleby11d7d8d9=728isdivisibleby13d8d9d10=289isdivisibleby17

Findthesumofall0to9pandigitalnumberswiththisproperty.

Problem44:Pentagonnumbers

Pentagonalnumbersaregeneratedbytheformula,Pn=n(3n1)/2.Thefirsttenpentagonalnumbersare:



1,5,12,22,35,51,70,92,117,145,...

ItcanbeseenthatP4+P7=22+70=92=P8.However,theirdifference,7022=48,isnotpentagonal.

Findthepairofpentagonalnumbers,PjandPk,forwhichtheirsumanddifferencearepentagonalandD=|PkPj|isminimisedwhatisthevalueofD?

Problem45:Triangular,pentagonal,andhexagonal

Triangle,pentagonal,andhexagonalnumbersaregeneratedbythefollowingformulae:

Triangle Tn=n(n+1)/2 1,3,6,10,15,...Pentagonal Pn=n(3n1)/2 1,5,12,22,35,...Hexagonal Hn=n(2n1) 1,6,15,28,45,...

ItcanbeverifiedthatT285=P165=H143=40755.

Findthenexttrianglenumberthatisalsopentagonalandhexagonal.

Problem46:Goldbach'sotherconjecture

ItwasproposedbyChristianGoldbachthateveryoddcompositenumbercanbewrittenasthesumofaprimeandtwiceasquare.

9=7+212

15=7+222

21=3+232

25=7+232

27=19+222

33=31+212

Itturnsoutthattheconjecturewasfalse.

Whatisthesmallestoddcompositethatcannotbewrittenasthesumofaprimeandtwiceasquare?

Problem47:Distinctprimesfactors

Thefirsttwoconsecutivenumberstohavetwodistinctprimefactorsare:

14=2715=35

Thefirstthreeconsecutivenumberstohavethreedistinctprimefactorsare:

644=2723645=3543646=21719.

Findthefirstfourconsecutiveintegerstohavefourdistinctprimefactors.Whatisthefirstofthesenumbers?

Problem48:Selfpowers

Theseries,11+22+33+...+1010=10405071317.

Findthelasttendigitsoftheseries,11+22+33+...+10001000.

Problem49:Primepermutations

Thearithmeticsequence,1487,4817,8147,inwhicheachofthetermsincreasesby3330,isunusualintwoways:(i)eachofthethreetermsareprime,and,(ii)eachofthe4digitnumbersarepermutationsofoneanother.



Therearenoarithmeticsequencesmadeupofthree1,2,or3digitprimes,exhibitingthisproperty,butthereisoneother4digitincreasingsequence.

What12digitnumberdoyouformbyconcatenatingthethreetermsinthissequence?

Problem50:Consecutiveprimesum

Theprime41,canbewrittenasthesumofsixconsecutiveprimes:

41=2+3+5+7+11+13

Thisisthelongestsumofconsecutiveprimesthataddstoaprimebelowonehundred.

Thelongestsumofconsecutiveprimesbelowonethousandthataddstoaprime,contains21terms,andisequalto953.

Whichprime,belowonemillion,canbewrittenasthesumofthemostconsecutiveprimes?

Problem51:Primedigitreplacements

Byreplacingthe1stdigitofthe2digitnumber*3,itturnsoutthatsixoftheninepossiblevalues:13,23,43,53,73,and83,areallprime.

Byreplacingthe3rdand4thdigitsof56**3withthesamedigit,this5digitnumberisthefirstexamplehavingsevenprimesamongthetengeneratednumbers,yieldingthefamily:56003,56113,56333,56443,56663,56773,and56993.Consequently56003,beingthefirstmemberofthisfamily,isthesmallestprimewiththisproperty.

Findthesmallestprimewhich,byreplacingpartofthenumber(notnecessarilyadjacentdigits)withthesamedigit,ispartofaneightprimevaluefamily.

Problem52:Permutedmultiples

Itcanbeseenthatthenumber,125874,anditsdouble,251748,containexactlythesamedigits,butinadifferentorder.

Findthesmallestpositiveinteger,x,suchthat2x,3x,4x,5x,and6x,containthesamedigits.

Problem53:Combinatoricselections

Thereareexactlytenwaysofselectingthreefromfive,12345:

123,124,125,134,135,145,234,235,245,and345

Incombinatorics,weusethenotation,5C3=10.

Ingeneral,

nCr=n!

r!(nr)! ,wherern,n!=n(n1)...321,and0!=1.

Itisnotuntiln=23,thatavalueexceedsonemillion:23C10=1144066.

Howmany,notnecessarilydistinct,valuesofnCr,for1n100,aregreaterthanonemillion?

Problem54:Pokerhands

Inthecardgamepoker,ahandconsistsoffivecardsandareranked,fromlowesttohighest,inthefollowingway:

HighCard:Highestvaluecard.OnePair:Twocardsofthesamevalue.TwoPairs:Twodifferentpairs.ThreeofaKind:Threecardsofthesamevalue.



Straight:Allcardsareconsecutivevalues.Flush:Allcardsofthesamesuit.FullHouse:Threeofakindandapair.FourofaKind:Fourcardsofthesamevalue.StraightFlush:Allcardsareconsecutivevaluesofsamesuit.RoyalFlush:Ten,Jack,Queen,King,Ace,insamesuit.

Thecardsarevaluedintheorder:2,3,4,5,6,7,8,9,10,Jack,Queen,King,Ace.

Iftwoplayershavethesamerankedhandsthentherankmadeupofthehighestvaluewinsforexample,apairofeightsbeatsapairoffives(seeexample1below).Butiftworankstie,forexample,bothplayershaveapairofqueens,thenhighestcardsineachhandarecompared(seeexample4below)ifthehighestcardstiethenthenexthighestcardsarecompared,andsoon.

Considerthefollowingfivehandsdealttotwoplayers:

Hand Player1 Player2 Winner1 5H5C6S7SKDPairofFives

2C3S8S8DTDPairofEights

Player2

2 5D8C9SJSACHighestcardAce 2C5C7D8SQHHighestcardQueen

Player1

3 2D9CASAHACThreeAces 3D6D7DTDQDFlushwithDiamonds

Player2

44D6S9HQHQC

PairofQueensHighestcardNine

3D6D7HQDQS

PairofQueensHighestcardSeven

Player1

52H2D4C4D4S

FullHouseWithThreeFours

3C3D3S9S9D

FullHousewithThreeThrees

Player1

Thefile,poker.txt,containsonethousandrandomhandsdealttotwoplayers.Eachlineofthefilecontainstencards(separatedbyasinglespace):thefirstfivearePlayer1'scardsandthelastfivearePlayer2'scards.Youcanassumethatallhandsarevalid(noinvalidcharactersorrepeatedcards),eachplayer'shandisinnospecificorder,andineachhandthereisaclearwinner.

HowmanyhandsdoesPlayer1win?

Problem55:Lychrelnumbers

Ifwetake47,reverseandadd,47+74=121,whichispalindromic.

Notallnumbersproducepalindromessoquickly.Forexample,

349+943=1292,1292+2921=42134213+3124=7337

Thatis,349tookthreeiterationstoarriveatapalindrome.

Althoughnoonehasprovedityet,itisthoughtthatsomenumbers,like196,neverproduceapalindrome.AnumberthatneverformsapalindromethroughthereverseandaddprocessiscalledaLychrelnumber.Duetothetheoreticalnatureofthesenumbers,andforthepurposeofthisproblem,weshallassumethatanumberisLychreluntilprovenotherwise.Inadditionyouaregiventhatforeverynumberbelowtenthousand,itwilleither(i)becomeapalindromeinlessthanfiftyiterations,or,(ii)noone,withallthecomputingpowerthatexists,hasmanagedsofartomapittoapalindrome.Infact,10677isthefirstnumbertobeshowntorequireoverfiftyiterationsbeforeproducingapalindrome:4668731596684224866951378664(53iterations,28digits).

Surprisingly,therearepalindromicnumbersthatarethemselvesLychrelnumbersthefirstexampleis4994.

HowmanyLychrelnumbersaretherebelowtenthousand?

NOTE:Wordingwasmodifiedslightlyon24April2007toemphasisethetheoreticalnatureofLychrelnumbers.

Problem56:Powerfuldigitsum

Agoogol(10100)isamassivenumber:onefollowedbyonehundredzeros100100isalmostunimaginablylarge:onefollowedbytwohundredzeros.Despitetheirsize,thesumofthedigitsineachnumberisonly1.

Consideringnaturalnumbersoftheform,ab,wherea,b



Problem57:Squarerootconvergents

Itispossibletoshowthatthesquarerootoftwocanbeexpressedasaninfinitecontinuedfraction.

2=1+1/(2+1/(2+1/(2+...)))=1.414213...

Byexpandingthisforthefirstfouriterations,weget:

1+1/2=3/2=1.51+1/(2+1/2)=7/5=1.41+1/(2+1/(2+1/2))=17/12=1.41666...1+1/(2+1/(2+1/(2+1/2)))=41/29=1.41379...

Thenextthreeexpansionsare99/70,239/169,and577/408,buttheeighthexpansion,1393/985,isthefirstexamplewherethenumberofdigitsinthenumeratorexceedsthenumberofdigitsinthedenominator.

Inthefirstonethousandexpansions,howmanyfractionscontainanumeratorwithmoredigitsthandenominator?

Problem58:Spiralprimes

Startingwith1andspirallinganticlockwiseinthefollowingway,asquarespiralwithsidelength7isformed.

37363534333231381716151413303918543122940196121128412078910274221222324252643444546474849

Itisinterestingtonotethattheoddsquaresliealongthebottomrightdiagonal,butwhatismoreinterestingisthat8outofthe13numberslyingalongbothdiagonalsareprimethatis,aratioof8/1362%.

Ifonecompletenewlayeriswrappedaroundthespiralabove,asquarespiralwithsidelength9willbeformed.Ifthisprocessiscontinued,whatisthesidelengthofthesquarespiralforwhichtheratioofprimesalongbothdiagonalsfirstfallsbelow10%?

Problem59:XORdecryption

EachcharacteronacomputerisassignedauniquecodeandthepreferredstandardisASCII(AmericanStandardCodeforInformationInterchange).Forexample,uppercaseA=65,asterisk(*)=42,andlowercasek=107.

Amodernencryptionmethodistotakeatextfile,convertthebytestoASCII,thenXOReachbytewithagivenvalue,takenfromasecretkey.TheadvantagewiththeXORfunctionisthatusingthesameencryptionkeyontheciphertext,restorestheplaintextforexample,65XOR42=107,then107XOR42=65.

Forunbreakableencryption,thekeyisthesamelengthastheplaintextmessage,andthekeyismadeupofrandombytes.Theuserwouldkeeptheencryptedmessageandtheencryptionkeyindifferentlocations,andwithoutboth"halves",itisimpossibletodecryptthemessage.

Unfortunately,thismethodisimpracticalformostusers,sothemodifiedmethodistouseapasswordasakey.Ifthepasswordisshorterthanthemessage,whichislikely,thekeyisrepeatedcyclicallythroughoutthemessage.Thebalanceforthismethodisusingasufficientlylongpasswordkeyforsecurity,butshortenoughtobememorable.

Yourtaskhasbeenmadeeasy,astheencryptionkeyconsistsofthreelowercasecharacters.Usingcipher.txt(rightclickand'SaveLink/TargetAs...'),afilecontainingtheencryptedASCIIcodes,andtheknowledgethattheplaintextmustcontaincommonEnglishwords,decryptthemessageandfindthesumoftheASCIIvaluesintheoriginaltext.

Problem60:Primepairsets

Theprimes3,7,109,and673,arequiteremarkable.Bytakinganytwoprimesandconcatenatingtheminanyordertheresultwillalwaysbeprime.Forexample,taking7and109,both7109and1097areprime.Thesumofthesefourprimes,792,representsthelowestsumforasetoffourprimeswiththisproperty.

Findthelowestsumforasetoffiveprimesforwhichanytwoprimesconcatenatetoproduceanotherprime.



Problem61:Cyclicalfiguratenumbers

Triangle,square,pentagonal,hexagonal,heptagonal,andoctagonalnumbersareallfigurate(polygonal)numbersandaregeneratedbythefollowingformulae:

Triangle P3,n=n(n+1)/2 1,3,6,10,15,...Square P4,n=n2 1,4,9,16,25,...Pentagonal P5,n=n(3n1)/2 1,5,12,22,35,...Hexagonal P6,n=n(2n1) 1,6,15,28,45,...Heptagonal P7,n=n(5n3)/2 1,7,18,34,55,...Octagonal P8,n=n(3n2) 1,8,21,40,65,...

Theorderedsetofthree4digitnumbers:8128,2882,8281,hasthreeinterestingproperties.

1. Thesetiscyclic,inthatthelasttwodigitsofeachnumberisthefirsttwodigitsofthenextnumber(includingthelastnumberwiththefirst).

2. Eachpolygonaltype:triangle(P3,127=8128),square(P4,91=8281),andpentagonal(P5,44=2882),isrepresentedbyadifferentnumberintheset.

3. Thisistheonlysetof4digitnumberswiththisproperty.

Findthesumoftheonlyorderedsetofsixcyclic4digitnumbersforwhicheachpolygonaltype:triangle,square,pentagonal,hexagonal,heptagonal,andoctagonal,isrepresentedbyadifferentnumberintheset.

Problem62:Cubicpermutations

Thecube,41063625(3453),canbepermutedtoproducetwoothercubes:56623104(3843)and66430125(4053).Infact,41063625isthesmallestcubewhichhasexactlythreepermutationsofitsdigitswhicharealsocube.

Findthesmallestcubeforwhichexactlyfivepermutationsofitsdigitsarecube.

Problem63:Powerfuldigitcounts

The5digitnumber,16807=75,isalsoafifthpower.Similarly,the9digitnumber,134217728=89,isaninthpower.

Howmanyndigitpositiveintegersexistwhicharealsoannthpower?

Problem64:Oddperiodsquareroots

Allsquarerootsareperiodicwhenwrittenascontinuedfractionsandcanbewrittenintheform:

N=a0+ 1 a1+ 1 a2+ 1 a3+...

Forexample,letusconsider23:

23=4+234=4+ 1 =4+ 1

1234 1+233

7

Ifwecontinuewewouldgetthefollowingexpansion:

23=4+ 1 1+ 1 3+ 1 1+ 1 8+...

Theprocesscanbesummarisedasfollows:

a0=4,1

234=23+4

7 =1+233

77 7(23+3) 233



a1=1,233= 14 =3+ 2

a2=3,2

233=2(23+3)

14 =1+234

7a3=1,

7234=

7(23+4)7 =8+234

a4=8,1

234=23+4

7 =1+233

7a5=1,

7233=

7(23+3)14 =3+

2332

a6=3,2

233=2(23+3)

14 =1+234

7a7=1,

7234=

7(23+4)7 =8+234

Itcanbeseenthatthesequenceisrepeating.Forconciseness,weusethenotation23=[4(1,3,1,8)],toindicatethattheblock(1,3,1,8)repeatsindefinitely.

Thefirsttencontinuedfractionrepresentationsof(irrational)squarerootsare:

2=[1(2)],period=13=[1(1,2)],period=25=[2(4)],period=16=[2(2,4)],period=27=[2(1,1,1,4)],period=48=[2(1,4)],period=210=[3(6)],period=111=[3(3,6)],period=212=[3(2,6)],period=213=[3(1,1,1,1,6)],period=5

Exactlyfourcontinuedfractions,forN13,haveanoddperiod.

HowmanycontinuedfractionsforN10000haveanoddperiod?

Problem65:Convergentsofe

Thesquarerootof2canbewrittenasaninfinitecontinuedfraction.

2=1+ 1 2+ 1 2+ 1 2+ 1 2+...

Theinfinitecontinuedfractioncanbewritten,2=[1(2)],(2)indicatesthat2repeatsadinfinitum.Inasimilarway,23=[4(1,3,1,8)].

Itturnsoutthatthesequenceofpartialvaluesofcontinuedfractionsforsquarerootsprovidethebestrationalapproximations.Letusconsidertheconvergentsfor2.

1+1=3/2 21+ 1 =7/5 2+1 21+ 1 =17/12 2+ 1 2+1 21+ 1 =41/29 2+ 1 2+ 1 2+1 2

Hencethesequenceofthefirsttenconvergentsfor2are:

1,3/2,7/5,17/12,41/29,99/70,239/169,577/408,1393/985,3363/2378,...

Whatismostsurprisingisthattheimportantmathematicalconstant,e=[21,2,1,1,4,1,1,6,1,...,1,2k,1,...].

Thefirsttentermsinthesequenceofconvergentsforeare:

2,3,8/3,11/4,19/7,87/32,106/39,193/71,1264/465,1457/536,...



Thesumofdigitsinthenumeratorofthe10thconvergentis1+4+5+7=17.

Findthesumofdigitsinthenumeratorofthe100thconvergentofthecontinuedfractionfore.

Problem66:Diophantineequation

ConsiderquadraticDiophantineequationsoftheform:

x2Dy2=1

Forexample,whenD=13,theminimalsolutioninxis6492131802=1.

ItcanbeassumedthattherearenosolutionsinpositiveintegerswhenDissquare.

ByfindingminimalsolutionsinxforD={2,3,5,6,7},weobtainthefollowing:

32222=122312=192542=152622=182732=1

Hence,byconsideringminimalsolutionsinxforD7,thelargestxisobtainedwhenD=5.

FindthevalueofD1000inminimalsolutionsofxforwhichthelargestvalueofxisobtained.

Problem67:MaximumpathsumII

Bystartingatthetopofthetrianglebelowandmovingtoadjacentnumbersontherowbelow,themaximumtotalfromtoptobottomis23.

374

2468593

Thatis,3+7+4+9=23.

Findthemaximumtotalfromtoptobottomintriangle.txt(rightclickand'SaveLink/TargetAs...'),a15Ktextfilecontainingatrianglewithonehundredrows.

NOTE:ThisisamuchmoredifficultversionofProblem18.Itisnotpossibletotryeveryroutetosolvethisproblem,asthereare299altogether!Ifyoucouldcheckonetrillion(1012)routeseveryseconditwouldtakeovertwentybillionyearstocheckthemall.Thereisanefficientalgorithmtosolveit.o)

Problem68:Magic5gonring

Considerthefollowing"magic"3gonring,filledwiththenumbers1to6,andeachlineaddingtonine.

Workingclockwise,andstartingfromthegroupofthreewiththenumericallylowestexternalnode(4,3,2inthisexample),eachsolutioncanbedescribeduniquely.Forexample,theabovesolutioncanbedescribedbytheset:4,3,26,2,15,1,3.

Itispossibletocompletetheringwithfourdifferenttotals:9,10,11,and12.Thereareeightsolutionsintotal.



Total SolutionSet9 4,2,35,3,16,1,29 4,3,26,2,15,1,310 2,3,54,5,16,1,310 2,5,36,3,14,1,511 1,4,63,6,25,2,411 1,6,45,4,23,2,612 1,5,62,6,43,4,512 1,6,53,5,42,4,6

Byconcatenatingeachgroupitispossibletoform9digitstringsthemaximumstringfora3gonringis432621513.

Usingthenumbers1to10,anddependingonarrangements,itispossibletoform16and17digitstrings.Whatisthemaximum16digitstringfora"magic"5gonring?

Problem69:Totientmaximum

Euler'sTotientfunction,(n)[sometimescalledthephifunction],isusedtodeterminethenumberofnumberslessthannwhicharerelativelyprimeton.Forexample,as1,2,4,5,7,and8,arealllessthannineandrelativelyprimetonine,(9)=6.

n RelativelyPrime (n) n/(n)2 1 1 23 1,2 2 1.54 1,3 2 25 1,2,3,4 4 1.256 1,5 2 37 1,2,3,4,5,6 6 1.1666...8 1,3,5,7 4 29 1,2,4,5,7,8 6 1.510 1,3,7,9 4 2.5

Itcanbeseenthatn=6producesamaximumn/(n)forn10.

Findthevalueofn1,000,000forwhichn/(n)isamaximum.

Problem70:Totientpermutation

Euler'sTotientfunction,(n)[sometimescalledthephifunction],isusedtodeterminethenumberofpositivenumberslessthanorequaltonwhicharerelativelyprimeton.Forexample,as1,2,4,5,7,and8,arealllessthannineandrelativelyprimetonine,(9)=6.Thenumber1isconsideredtoberelativelyprimetoeverypositivenumber,so(1)=1.

Interestingly,(87109)=79180,anditcanbeseenthat87109isapermutationof79180.

Findthevalueofn,1



Considerthefraction,n/d,wherenanddarepositiveintegers.Ifn



30cm:(5,12,13)36cm:(9,12,15)40cm:(8,15,17)48cm:(12,16,20)

Incontrast,somelengthsofwire,like20cm,cannotbebenttoformanintegersidedrightangletriangle,andotherlengthsallowmorethanonesolutiontobefoundforexample,using120cmitispossibletoformexactlythreedifferentintegersidedrightangletriangles.

120cm:(30,40,50),(20,48,52),(24,45,51)

GiventhatListhelengthofthewire,forhowmanyvaluesofL1,500,000canexactlyoneintegersidedrightangletrianglebeformed?

Problem76:Countingsummations

Itispossibletowritefiveasasuminexactlysixdifferentways:

4+13+23+1+12+2+12+1+1+11+1+1+1+1

Howmanydifferentwayscanonehundredbewrittenasasumofatleasttwopositiveintegers?

Problem77:Primesummations

Itispossibletowritetenasthesumofprimesinexactlyfivedifferentways:

7+35+55+3+23+3+2+22+2+2+2+2

Whatisthefirstvaluewhichcanbewrittenasthesumofprimesinoverfivethousanddifferentways?

Problem78:Coinpartitions

Letp(n)representthenumberofdifferentwaysinwhichncoinscanbeseparatedintopiles.Forexample,fivecoinscanbeseparatedintopilesinexactlysevendifferentways,sop(5)=7.

OOOOO

OOOOO

OOOOO

OOOOO

OOOOO

OOOOO

OOOOO

Findtheleastvalueofnforwhichp(n)isdivisiblebyonemillion.

Problem79:Passcodederivation

Acommonsecuritymethodusedforonlinebankingistoasktheuserforthreerandomcharactersfromapasscode.Forexample,ifthepasscodewas531278,theymayaskforthe2nd,3rd,and5thcharacterstheexpectedreplywouldbe:317.

Thetextfile,keylog.txt,containsfiftysuccessfulloginattempts.



Giventhatthethreecharactersarealwaysaskedforinorder,analysethefilesoastodeterminetheshortestpossiblesecretpasscodeofunknownlength.

Problem80:Squarerootdigitalexpansion

Itiswellknownthatifthesquarerootofanaturalnumberisnotaninteger,thenitisirrational.Thedecimalexpansionofsuchsquarerootsisinfinitewithoutanyrepeatingpatternatall.

Thesquarerootoftwois1.41421356237309504880...,andthedigitalsumofthefirstonehundreddecimaldigitsis475.

Forthefirstonehundrednaturalnumbers,findthetotalofthedigitalsumsofthefirstonehundreddecimaldigitsforalltheirrationalsquareroots.

Problem81:Pathsum:twoways

Inthe5by5matrixbelow,theminimalpathsumfromthetoplefttothebottomright,byonlymovingtotherightanddown,isindicatedinboldredandisequalto2427.

Findtheminimalpathsum,inmatrix.txt(rightclickand"SaveLink/TargetAs..."),a31Ktextfilecontaininga80by80matrix,fromthetoplefttothebottomrightbyonlymovingrightanddown.

Problem82:Pathsum:threeways

NOTE:ThisproblemisamorechallengingversionofProblem81.

Theminimalpathsuminthe5by5matrixbelow,bystartinginanycellintheleftcolumnandfinishinginanycellintherightcolumn,andonlymovingup,down,andright,isindicatedinredandboldthesumisequalto994.

Findtheminimalpathsum,inmatrix.txt(rightclickand"SaveLink/TargetAs..."),a31Ktextfilecontaininga80by80matrix,fromtheleftcolumntotherightcolumn.

Problem83:Pathsum:fourways

NOTE:ThisproblemisasignificantlymorechallengingversionofProblem81.

Inthe5by5matrixbelow,theminimalpathsumfromthetoplefttothebottomright,bymovingleft,right,up,anddown,isindicatedinboldredandisequalto2297.

Findtheminimalpathsum,inmatrix.txt(rightclickand"SaveLink/TargetAs..."),a31Ktextfilecontaininga80by80matrix,fromthetoplefttothebottomrightbymovingleft,right,up,anddown.



Problem84:Monopolyodds

Inthegame,Monopoly,thestandardboardissetupinthefollowingway:

GO A1 CC1 A2 T1 R1 B1 CH1 B2 B3 JAIL

H2 C1

T2 U1

H1 C2

CH3 C3

R4 R2

G3 D1

CC3 CC2

G2 D2

G1 D3

G2J F3 U2 F2 F1 R3 E3 E2 CH2 E1 FP

AplayerstartsontheGOsquareandaddsthescoresontwo6sideddicetodeterminethenumberofsquarestheyadvanceinaclockwisedirection.Withoutanyfurtherruleswewouldexpecttovisiteachsquarewithequalprobability:2.5%.However,landingonG2J(GoToJail),CC(communitychest),andCH(chance)changesthisdistribution.

InadditiontoG2J,andonecardfromeachofCCandCH,thatorderstheplayertogodirectlytojail,ifaplayerrollsthreeconsecutivedoubles,theydonotadvancetheresultoftheir3rdroll.Insteadtheyproceeddirectlytojail.

Atthebeginningofthegame,theCCandCHcardsareshuffled.WhenaplayerlandsonCCorCHtheytakeacardfromthetopoftherespectivepileand,afterfollowingtheinstructions,itisreturnedtothebottomofthepile.Therearesixteencardsineachpile,butforthepurposeofthisproblemweareonlyconcernedwithcardsthatorderamovementanyinstructionnotconcernedwithmovementwillbeignoredandtheplayerwillremainontheCC/CHsquare.

CommunityChest(2/16cards):1. AdvancetoGO2. GotoJAIL

Chance(10/16cards):1. AdvancetoGO2. GotoJAIL3. GotoC14. GotoE35. GotoH26. GotoR17. GotonextR(railwaycompany)8. GotonextR9. GotonextU(utilitycompany)

10. Goback3squares.

Theheartofthisproblemconcernsthelikelihoodofvisitingaparticularsquare.Thatis,theprobabilityoffinishingatthatsquareafteraroll.Forthisreasonitshouldbeclearthat,withtheexceptionofG2Jforwhichtheprobabilityoffinishingonitiszero,theCHsquareswillhavethelowestprobabilities,as5/8requestamovementtoanothersquare,anditisthefinalsquarethattheplayerfinishesatoneachrollthatweareinterestedin.Weshallmakenodistinctionbetween"JustVisiting"andbeingsenttoJAIL,andweshallalsoignoretheruleaboutrequiringadoubleto"getoutofjail",assumingthattheypaytogetoutontheirnextturn.

BystartingatGOandnumberingthesquaressequentiallyfrom00to39wecanconcatenatethesetwodigitnumberstoproducestringsthatcorrespondwithsetsofsquares.

Statisticallyitcanbeshownthatthethreemostpopularsquares,inorder,areJAIL(6.24%)=Square10,E3(3.18%)=Square24,andGO(3.09%)=Square00.Sothesethreemostpopularsquarescanbelistedwiththesixdigitmodalstring:102400.

If,insteadofusingtwo6sideddice,two4sideddiceareused,findthesixdigitmodalstring.

Problem85:Countingrectangles

Bycountingcarefullyitcanbeseenthatarectangulargridmeasuring3by2containseighteenrectangles:



Althoughthereexistsnorectangulargridthatcontainsexactlytwomillionrectangles,findtheareaofthegridwiththenearestsolution.

Problem86:Cuboidroute

Aspider,S,sitsinonecornerofacuboidroom,measuring6by5by3,andafly,F,sitsintheoppositecorner.Bytravellingonthesurfacesoftheroomtheshortest"straightline"distancefromStoFis10andthepathisshownonthediagram.

However,thereareuptothree"shortest"pathcandidatesforanygivencuboidandtheshortestroutedoesn'talwayshaveintegerlength.

Itcanbeshownthatthereareexactly2060distinctcuboids,ignoringrotations,withintegerdimensions,uptoamaximumsizeofMbyMbyM,forwhichtheshortestroutehasintegerlengthwhenM=100.ThisistheleastvalueofMforwhichthenumberofsolutionsfirstexceedstwothousandthenumberofsolutionswhenM=99is1975.

FindtheleastvalueofMsuchthatthenumberofsolutionsfirstexceedsonemillion.

Problem87:Primepowertriples

Thesmallestnumberexpressibleasthesumofaprimesquare,primecube,andprimefourthpoweris28.Infact,thereareexactlyfournumbersbelowfiftythatcanbeexpressedinsuchaway:

28=22+23+24

33=32+23+24

49=52+23+24

47=22+33+24

Howmanynumbersbelowfiftymillioncanbeexpressedasthesumofaprimesquare,primecube,andprimefourthpower?

Problem88:Productsumnumbers

Anaturalnumber,N,thatcanbewrittenasthesumandproductofagivensetofatleasttwonaturalnumbers,{a1,a2,...,ak}iscalledaproductsumnumber:N=a1+a2+...+ak=a1a2...ak.

Forexample,6=1+2+3=123.

Foragivensetofsize,k,weshallcallthesmallestNwiththispropertyaminimalproductsumnumber.Theminimalproductsumnumbersforsetsofsize,k=2,3,4,5,and6areasfollows.

k=2:4=22=2+2k=3:6=123=1+2+3k=4:8=1124=1+1+2+4k=5:8=11222=1+1+2+2+2k=6:12=111126=1+1+1+1+2+6



Hencefor2k6,thesumofalltheminimalproductsumnumbersis4+6+8+12=30notethat8isonlycountedonceinthesum.

Infact,asthecompletesetofminimalproductsumnumbersfor2k12is{4,6,8,12,15,16},thesumis61.

Whatisthesumofalltheminimalproductsumnumbersfor2k12000?

Problem89:Romannumerals

ForanumberwritteninRomannumeralstobeconsideredvalidtherearebasicruleswhichmustbefollowed.Eventhoughtherulesallowsomenumberstobeexpressedinmorethanonewaythereisalwaysa"best"wayofwritingaparticularnumber.

Forexample,itwouldappearthatthereareatleastsixwaysofwritingthenumbersixteen:

IIIIIIIIIIIIIIIIVIIIIIIIIIIIVVIIIIIIXIIIIIIVVVIXVI

However,accordingtotherulesonlyXIIIIIIandXVIarevalid,andthelastexampleisconsideredtobethemostefficient,asitusestheleastnumberofnumerals.

The11Ktextfile,roman.txt(rightclickand'SaveLink/TargetAs...'),containsonethousandnumberswritteninvalid,butnotnecessarilyminimal,RomannumeralsseeAbout...RomanNumeralsforthedefinitiverulesforthisproblem.

Findthenumberofcharacterssavedbywritingeachoftheseintheirminimalform.

Note:YoucanassumethatalltheRomannumeralsinthefilecontainnomorethanfourconsecutiveidenticalunits.

Problem90:Cubedigitpairs

Eachofthesixfacesonacubehasadifferentdigit(0to9)writtenonitthesameisdonetoasecondcube.Byplacingthetwocubessidebysideindifferentpositionswecanformavarietyof2digitnumbers.

Forexample,thesquarenumber64couldbeformed:

Infact,bycarefullychoosingthedigitsonbothcubesitispossibletodisplayallofthesquarenumbersbelowonehundred:01,04,09,16,25,36,49,64,and81.

Forexample,onewaythiscanbeachievedisbyplacing{0,5,6,7,8,9}ononecubeand{1,2,3,4,8,9}ontheothercube.

However,forthisproblemweshallallowthe6or9tobeturnedupsidedownsothatanarrangementlike{0,5,6,7,8,9}and{1,2,3,4,6,7}allowsforallninesquarenumberstobedisplayedotherwiseitwouldbeimpossibletoobtain09.

Indeterminingadistinctarrangementweareinterestedinthedigitsoneachcube,nottheorder.

{1,2,3,4,5,6}isequivalentto{3,6,4,1,2,5}{1,2,3,4,5,6}isdistinctfrom{1,2,3,4,5,9}

Butbecauseweareallowing6and9tobereversed,thetwodistinctsetsinthelastexamplebothrepresenttheextendedset{1,2,3,4,5,6,9}forthepurposeofforming2digitnumbers.

Howmanydistinctarrangementsofthetwocubesallowforallofthesquarenumberstobedisplayed?

Problem91:Righttriangleswithintegercoordinates

ThepointsP(x1,y1)andQ(x2,y2)areplottedatintegercoordinatesandarejoinedtotheorigin,O(0,0),to



formOPQ.

Thereareexactlyfourteentrianglescontainingarightanglethatcanbeformedwheneachcoordinateliesbetween0and2inclusivethatis,0x1,y1,x2,y22.

Giventhat0x1,y1,x2,y250,howmanyrighttrianglescanbeformed?

Problem92:Squaredigitchains

Anumberchainiscreatedbycontinuouslyaddingthesquareofthedigitsinanumbertoformanewnumberuntilithasbeenseenbefore.

Forexample,

443213101185891454220416375889

Thereforeanychainthatarrivesat1or89willbecomestuckinanendlessloop.WhatismostamazingisthatEVERYstartingnumberwilleventuallyarriveat1or89.

Howmanystartingnumbersbelowtenmillionwillarriveat89?

Problem93:Arithmeticexpressions

Byusingeachofthedigitsfromtheset,{1,2,3,4},exactlyonce,andmakinguseofthefourarithmeticoperations(+,,*,/)andbrackets/parentheses,itispossibletoformdifferentpositiveintegertargets.

Forexample,

8=(4*(1+3))/214=4*(3+1/2)19=4*(2+3)136=3*4*(2+1)

Notethatconcatenationsofthedigits,like12+34,arenotallowed.

Usingtheset,{1,2,3,4},itispossibletoobtainthirtyonedifferenttargetnumbersofwhich36isthemaximum,andeachofthenumbers1to28canbeobtainedbeforeencounteringthefirstnonexpressiblenumber.

Findthesetoffourdistinctdigits,a



Problem94:Almostequilateraltriangles

Itiseasilyprovedthatnoequilateraltriangleexistswithintegrallengthsidesandintegralarea.However,thealmostequilateraltriangle556hasanareaof12squareunits.

Weshalldefineanalmostequilateraltriangletobeatriangleforwhichtwosidesareequalandthethirddiffersbynomorethanoneunit.

Findthesumoftheperimetersofallalmostequilateraltriangleswithintegralsidelengthsandareaandwhoseperimetersdonotexceedonebillion(1,000,000,000).

Problem95:Amicablechains

Theproperdivisorsofanumberareallthedivisorsexcludingthenumberitself.Forexample,theproperdivisorsof28are1,2,4,7,and14.Asthesumofthesedivisorsisequalto28,wecallitaperfectnumber.

Interestinglythesumoftheproperdivisorsof220is284andthesumoftheproperdivisorsof284is220,formingachainoftwonumbers.Forthisreason,220and284arecalledanamicablepair.

Perhapslesswellknownarelongerchains.Forexample,startingwith12496,weformachainoffivenumbers:

1249614288154721453614264(12496...)

Sincethischainreturnstoitsstartingpoint,itiscalledanamicablechain.

Findthesmallestmemberofthelongestamicablechainwithnoelementexceedingonemillion.

Problem96:SuDoku

SuDoku(Japanesemeaningnumberplace)isthenamegiventoapopularpuzzleconcept.Itsoriginisunclear,butcreditmustbeattributedtoLeonhardEulerwhoinventedasimilar,andmuchmoredifficult,puzzleideacalledLatinSquares.TheobjectiveofSuDokupuzzles,however,istoreplacetheblanks(orzeros)ina9by9gridinsuchthateachrow,column,and3by3boxcontainseachofthedigits1to9.Belowisanexampleofatypicalstartingpuzzlegridanditssolutiongrid.

003900001

020305806

600001400

008700006

102000708

900008200

002800005

609203010

500009300

483967251

921345876

657821493

548729136

132564798

976138245

372814695

689253417

514769382

AwellconstructedSuDokupuzzlehasauniquesolutionandcanbesolvedbylogic,althoughitmaybenecessarytoemploy"guessandtest"methodsinordertoeliminateoptions(thereismuchcontestedopinionoverthis).Thecomplexityofthesearchdeterminesthedifficultyofthepuzzletheexampleaboveisconsideredeasybecauseitcanbesolvedbystraightforwarddirectdeduction.

The6Ktextfile,sudoku.txt(rightclickand'SaveLink/TargetAs...'),containsfiftydifferentSuDokupuzzlesrangingindifficulty,butallwithuniquesolutions(thefirstpuzzleinthefileistheexampleabove).

Bysolvingallfiftypuzzlesfindthesumofthe3digitnumbersfoundinthetopleftcornerofeachsolutiongridforexample,483isthe3digitnumberfoundinthetopleftcornerofthesolutiongridabove.

Problem97:LargenonMersenneprime

Thefirstknownprimefoundtoexceedonemilliondigitswasdiscoveredin1999,andisaMersenneprimeoftheform269725931itcontainsexactly2,098,960digits.SubsequentlyotherMersenneprimes,oftheform2p1,havebeenfoundwhichcontainmoredigits.

However,in2004therewasfoundamassivenonMersenneprimewhichcontains2,357,207digits:2843327830457+1.

Findthelasttendigitsofthisprimenumber.



Problem98:Anagramicsquares

ByreplacingeachofthelettersinthewordCAREwith1,2,9,and6respectively,weformasquarenumber:1296=362.Whatisremarkableisthat,byusingthesamedigitalsubstitutions,theanagram,RACE,alsoformsasquarenumber:9216=962.WeshallcallCARE(andRACE)asquareanagramwordpairandspecifyfurtherthatleadingzeroesarenotpermitted,neithermayadifferentletterhavethesamedigitalvalueasanotherletter.

Usingwords.txt(rightclickand'SaveLink/TargetAs...'),a16KtextfilecontainingnearlytwothousandcommonEnglishwords,findallthesquareanagramwordpairs(apalindromicwordisNOTconsideredtobeananagramofitself).

Whatisthelargestsquarenumberformedbyanymemberofsuchapair?

Problem99:Largestexponential

Comparingtwonumberswritteninindexformlike211and37isnotdifficult,asanycalculatorwouldconfirmthat211=2048519432525806wouldbemuchmoredifficult,asbothnumberscontainoverthreemilliondigits.

Usingbase_exp.txt(rightclickand'SaveLink/TargetAs...'),a22Ktextfilecontainingonethousandlineswithabase/exponentpaironeachline,determinewhichlinenumberhasthegreatestnumericalvalue.

Problem100:Arrangedprobability

Ifaboxcontainstwentyonecoloureddiscs,composedoffifteenbluediscsandsixreddiscs,andtwodiscsweretakenatrandom,itcanbeseenthattheprobabilityoftakingtwobluediscs,P(BB)=(15/21)(14/20)=1/2.

Thenextsucharrangement,forwhichthereisexactly50%chanceoftakingtwobluediscsatrandom,isaboxcontainingeightyfivebluediscsandthirtyfivereddiscs.

Byfindingthefirstarrangementtocontainover1012=1,000,000,000,000discsintotal,determinethenumberofbluediscsthattheboxwouldcontain.

Problem101:Optimumpolynomial

Ifwearepresentedwiththefirstktermsofasequenceitisimpossibletosaywithcertaintythevalueofthenextterm,asthereareinfinitelymanypolynomialfunctionsthatcanmodelthesequence.

Asanexample,letusconsiderthesequenceofcubenumbers.Thisisdefinedbythegeneratingfunction,un=n3:1,8,27,64,125,216,...

Supposewewereonlygiventhefirsttwotermsofthissequence.Workingontheprinciplethat"simpleisbest"weshouldassumealinearrelationshipandpredictthenexttermtobe15(commondifference7).Evenifwewerepresentedwiththefirstthreeterms,bythesameprincipleofsimplicity,aquadraticrelationshipshouldbeassumed.

WeshalldefineOP(k,n)tobethenthtermoftheoptimumpolynomialgeneratingfunctionforthefirstktermsofasequence.ItshouldbeclearthatOP(k,n)willaccuratelygeneratethetermsofthesequencefornk,andpotentiallythefirstincorrectterm(FIT)willbeOP(k,k+1)inwhichcaseweshallcallitabadOP(BOP).

Asabasis,ifwewereonlygiventhefirsttermofsequence,itwouldbemostsensibletoassumeconstancythatis,forn2,OP(1,n)=u1.

HenceweobtainthefollowingOPsforthecubicsequence:

OP(1,n)=1 1,1,1,1,...

NOTE:Allanagramsformedmustbecontainedinthegiventextfile.

NOTE:Thefirsttwolinesinthefilerepresentthenumbersintheexamplegivenabove.



OP(2,n)=7n6 1,8,15,...OP(3,n)=6n211n+6 1,8,27,58,...OP(4,n)=n3 1,8,27,64,125,...

ClearlynoBOPsexistfork4.

ByconsideringthesumofFITsgeneratedbytheBOPs(indicatedinredabove),weobtain1+15+58=74.

Considerthefollowingtenthdegreepolynomialgeneratingfunction:

un=1n+n2n3+n4n5+n6n7+n8n9+n10

FindthesumofFITsfortheBOPs.

Problem102:Trianglecontainment

ThreedistinctpointsareplottedatrandomonaCartesianplane,forwhich1000x,y1000,suchthatatriangleisformed.

Considerthefollowingtwotriangles:

A(340,495),B(153,910),C(835,947)

X(175,41),Y(421,714),Z(574,645)

ItcanbeverifiedthattriangleABCcontainstheorigin,whereastriangleXYZdoesnot.

Usingtriangles.txt(rightclickand'SaveLink/TargetAs...'),a27Ktextfilecontainingthecoordinatesofonethousand"random"triangles,findthenumberoftrianglesforwhichtheinteriorcontainstheorigin.

Problem103:Specialsubsetsums:optimum

LetS(A)representthesumofelementsinsetAofsizen.Weshallcallitaspecialsumsetifforanytwononemptydisjointsubsets,BandC,thefollowingpropertiesaretrue:

i. S(B)S(C)thatis,sumsofsubsetscannotbeequal.ii. IfBcontainsmoreelementsthanCthenS(B)>S(C).

IfS(A)isminimisedforagivenn,weshallcallitanoptimumspecialsumset.Thefirstfiveoptimumspecialsumsetsaregivenbelow.

n=1:{1}n=2:{1,2}n=3:{2,3,4}n=4:{3,5,6,7}n=5:{6,9,11,12,13}

Itseemsthatforagivenoptimumset,A={a1,a2,...,an},thenextoptimumsetisoftheformB={b,a1+b,a2+b,...,an+b},wherebisthe"middle"elementonthepreviousrow.

Byapplyingthis"rule"wewouldexpecttheoptimumsetforn=6tobeA={11,17,20,22,23,24},withS(A)=117.However,thisisnottheoptimumset,aswehavemerelyappliedanalgorithmtoprovideanearoptimumset.Theoptimumsetforn=6isA={11,18,19,20,22,25},withS(A)=115andcorrespondingsetstring:111819202225.

GiventhatAisanoptimumspecialsumsetforn=7,finditssetstring.

NOTE:ThisproblemisrelatedtoProblem105andProblem106.

Problem104:PandigitalFibonacciends

TheFibonaccisequenceisdefinedbytherecurrencerelation:

Fn=Fn1+Fn2,whereF1=1andF2=1.

ItturnsoutthatF541,whichcontains113digits,isthefirstFibonaccinumberforwhichthelastninedigitsare19pandigital(containallthedigits1to9,butnotnecessarilyinorder).AndF2749,whichcontains575

NOTE:Thefirsttwoexamplesinthefilerepresentthetrianglesintheexamplegivenabove.



digits,isthefirstFibonaccinumberforwhichthefirstninedigitsare19pandigital.

GiventhatFkisthefirstFibonaccinumberforwhichthefirstninedigitsANDthelastninedigitsare19pandigital,findk.

Problem105:Specialsubsetsums:testing



Forexample,{81,88,75,42,87,84,86,65}isnotaspecialsumsetbecause65+87+88=75+81+84,whereas{157,150,164,119,79,159,161,139,158}satisfiesbothrulesforallpossiblesubsetpaircombinationsandS(A)=1286.

Usingsets.txt(rightclickand"SaveLink/TargetAs..."),a4Ktextfilewithonehundredsetscontainingseventotwelveelements(thetwoexamplesgivenabovearethefirsttwosetsinthefile),identifyallthespecialsumsets,A1,A2,...,Ak,andfindthevalueofS(A1)+S(A2)+...+S(Ak).


Problem106:Specialsubsetsums:metatesting



Forthisproblemweshallassumethatagivensetcontainsnstrictlyincreasingelementsanditalreadysatisfiesthesecondrule.

Surprisingly,outofthe25possiblesubsetpairsthatcanbeobtainedfromasetforwhichn=4,only1ofthesepairsneedtobetestedforequality(firstrule).Similarly,whenn=7,only70outofthe966subsetpairsneedtobetested.

Forn=12,howmanyofthe261625subsetpairsthatcanbeobtainedneedtobetestedforequality?


Problem107:Minimalnetwork

Thefollowingundirectednetworkconsistsofsevenverticesandtwelveedgeswithatotalweightof243.

Thesamenetworkcanberepresentedbythematrixbelow.

A B C D E F G

A 16 12 21

B 16 17 20

C 12 28 31

D 21 17 28 18 19 23



E 20 18 11

F 31 19 27

G 23 11 27

However,itispossibletooptimisethenetworkbyremovingsomeedgesandstillensurethatallpointsonthenetworkremainconnected.Thenetworkwhichachievesthemaximumsavingisshownbelow.Ithasaweightof93,representingasavingof24393=150fromtheoriginalnetwork.

Usingnetwork.txt(rightclickand'SaveLink/TargetAs...'),a6Ktextfilecontaininganetworkwithfortyvertices,andgiveninmatrixform,findthemaximumsavingwhichcanbeachievedbyremovingredundantedgeswhilstensuringthatthenetworkremainsconnected.

Problem108:DiophantinereciprocalsI

Inthefollowingequationx,y,andnarepositiveintegers.

1x +

1y =

1n

Forn=4thereareexactlythreedistinctsolutions:

15 +

120 =

14

16 +

112 =

14

18 +

18 =

14

Whatistheleastvalueofnforwhichthenumberofdistinctsolutionsexceedsonethousand?

NOTE:ThisproblemisaneasierversionofProblem110itisstronglyadvisedthatyousolvethisonefirst.

Problem109:Darts

Inthegameofdartsaplayerthrowsthreedartsatatargetboardwhichissplitintotwentyequalsizedsectionsnumberedonetotwenty.



Thescoreofadartisdeterminedbythenumberoftheregionthatthedartlandsin.Adartlandingoutsidethered/greenouterringscoreszero.Theblackandcreamregionsinsidethisringrepresentsinglescores.However,thered/greenouterringandmiddleringscoredoubleandtreblescoresrespectively.

Atthecentreoftheboardaretwoconcentriccirclescalledthebullregion,orbullseye.Theouterbullisworth25pointsandtheinnerbullisadouble,worth50points.

Therearemanyvariationsofrulesbutinthemostpopulargametheplayerswillbeginwithascore301or501andthefirstplayertoreducetheirrunningtotaltozeroisawinner.However,itisnormaltoplaya"doublesout"system,whichmeansthattheplayermustlandadouble(includingthedoublebullseyeatthecentreoftheboard)ontheirfinaldarttowinanyotherdartthatwouldreducetheirrunningtotaltooneorlowermeansthescoreforthatsetofthreedartsis"bust".

Whenaplayerisabletofinishontheircurrentscoreitiscalleda"checkout"andthehighestcheckoutis170:T20T20D25(twotreble20sanddoublebull).

Thereareexactlyelevendistinctwaystocheckoutonascoreof6:

D3 D1 D2 S2 D2 D2 D1 S4 D1 S1 S1 D2S1 T1 D1S1 S3 D1D1 D1 D1D1 S2 D1S2 S2 D1

NotethatD1D2isconsidereddifferenttoD2D1astheyfinishondifferentdoubles.However,thecombinationS1T1D1isconsideredthesameasT1S1D1.

Inadditionweshallnotincludemissesinconsideringcombinationsforexample,D3isthesameas0D3and00D3.

Incrediblythereare42336distinctwaysofcheckingoutintotal.

Howmanydistinctwayscanaplayercheckoutwithascorelessthan100?

Problem110:DiophantinereciprocalsII

Inthefollowingequationx,y,andnarepositiveintegers.

1x +

1y =

1n



Itcanbeverifiedthatwhenn=1260thereare113distinctsolutionsandthisistheleastvalueofnforwhichthetotalnumberofdistinctsolutionsexceedsonehundred.

Whatistheleastvalueofnforwhichthenumberofdistinctsolutionsexceedsfourmillion?

NOTE:ThisproblemisamuchmoredifficultversionofProblem108andasitiswellbeyondthelimitationsofabruteforceapproachitrequiresacleverimplementation.

Problem111:Primeswithruns

Considering4digitprimescontainingrepeateddigitsitisclearthattheycannotallbethesame:1111isdivisibleby11,2222isdivisibleby22,andsoon.Buttherearenine4digitprimescontainingthreeones:

1117,1151,1171,1181,1511,1811,2111,4111,8111

WeshallsaythatM(n,d)representsthemaximumnumberofrepeateddigitsforanndigitprimewheredistherepeateddigit,N(n,d)representsthenumberofsuchprimes,andS(n,d)representsthesumoftheseprimes.

SoM(4,1)=3isthemaximumnumberofrepeateddigitsfora4digitprimewhereoneistherepeateddigit,thereareN(4,1)=9suchprimes,andthesumoftheseprimesisS(4,1)=22275.Itturnsoutthatford=0,itisonlypossibletohaveM(4,0)=2repeateddigits,butthereareN(4,0)=13suchcases.

Inthesamewayweobtainthefollowingresultsfor4digitprimes.

Digit,d M(4,d) N(4,d) S(4,d)

0 2 13 67061

1 3 9 22275

2 3 1 2221

3 3 12 46214

4 3 2 8888

5 3 1 5557

6 3 1 6661

7 3 9 57863

8 3 1 8887

9 3 7 48073

Ford=0to9,thesumofallS(4,d)is273700.

FindthesumofallS(10,d).

Problem112:Bouncynumbers

Workingfromlefttorightifnodigitisexceededbythedigittoitsleftitiscalledanincreasingnumberforexample,134468.

Similarlyifnodigitisexceededbythedigittoitsrightitiscalledadecreasingnumberforexample,66420.

Weshallcallapositiveintegerthatisneitherincreasingnordecreasinga"bouncy"numberforexample,155349.

Clearlytherecannotbeanybouncynumbersbelowonehundred,butjustoverhalfofthenumbersbelowonethousand(525)arebouncy.Infact,theleastnumberforwhichtheproportionofbouncynumbersfirstreaches50%is538.

Surprisingly,bouncynumbersbecomemoreandmorecommonandbythetimewereach21780theproportionofbouncynumbersisequalto90%.

Findtheleastnumberforwhichtheproportionofbouncynumbersisexactly99%.

Problem113:Nonbouncynumbers

Workingfromlefttorightifnodigitisexceededbythedigittoitsleftitiscalledanincreasingnumberforexample,134468.



Similarlyifnodigitisexceededbythedigittoitsrightitiscalledadecreasingnumberforexample,66420.

Weshallcallapositiveintegerthatisneitherincreasingnordecreasinga"bouncy"numberforexample,155349.

Asnincreases,theproportionofbouncynumbersbelownincreasessuchthatthereareonly12951numbersbelowonemillionthatarenotbouncyandonly277032nonbouncynumbersbelow1010.

Howmanynumbersbelowagoogol(10100)arenotbouncy?

Problem114:CountingblockcombinationsI

Arowmeasuringsevenunitsinlengthhasredblockswithaminimumlengthofthreeunitsplacedonit,suchthatanytworedblocks(whichareallowedtobedifferentlengths)areseparatedbyatleastoneblacksquare.Thereareexactlyseventeenwaysofdoingthis.

Howmanywayscanarowmeasuringfiftyunitsinlengthbefilled?

NOTE:Althoughtheexampleabovedoesnotlenditselftothepossibility,ingeneralitispermittedtomixblocksizes.Forexample,onarowmeasuringeightunitsinlengthyoucouldusered(3),black(1),andred(4).

Problem115:CountingblockcombinationsII

NOTE:ThisisamoredifficultversionofProblem114.

Arowmeasuringnunitsinlengthhasredblockswithaminimumlengthofmunitsplacedonit,suchthatanytworedblocks(whichareallowedtobedifferentlengths)areseparatedbyatleastoneblacksquare.

Letthefillcountfunction,F(m,n),representthenumberofwaysthatarowcanbefilled.

Forexample,F(3,29)=673135andF(3,30)=1089155.

Thatis,form=3,itcanbeseenthatn=30isthesmallestvalueforwhichthefillcountfunctionfirstexceedsonemillion.

Inthesameway,form=10,itcanbeverifiedthatF(10,56)=880711andF(10,57)=1148904,son=57istheleastvalueforwhichthefillcountfunctionfirstexceedsonemillion.

Form=50,findtheleastvalueofnforwhichthefillcountfunctionfirstexceedsonemillion.

Problem116:Red,greenorbluetiles

Arowoffiveblacksquaretilesistohaveanumberofitstilesreplacedwithcolouredoblongtileschosenfromred(lengthtwo),green(lengththree),orblue(lengthfour).

Ifredtilesarechosenthereareexactlysevenwaysthiscanbedone.

Ifgreentilesarechosentherearethreeways.



Andifbluetilesarechosentherearetwoways.

Assumingthatcolourscannotbemixedthereare7+3+2=12waysofreplacingtheblacktilesinarowmeasuringfiveunitsinlength.

Howmanydifferentwayscantheblacktilesinarowmeasuringfiftyunitsinlengthbereplacedifcolourscannotbemixedandatleastonecolouredtilemustbeused?

NOTE:ThisisrelatedtoProblem117.

Problem117:Red,green,andbluetiles

Usingacombinationofblacksquaretilesandoblongtileschosenfrom:redtilesmeasuringtwounits,greentilesmeasuringthreeunits,andbluetilesmeasuringfourunits,itispossibletotilearowmeasuringfiveunitsinlengthinexactlyfifteendifferentways.

Howmanywayscanarowmeasuringfiftyunitsinlengthbetiled?

NOTE:ThisisrelatedtoProblem116.

Problem118:Pandigitalprimesets

Usingallofthedigits1through9andconcatenatingthemfreelytoformdecimalintegers,differentsetscanbeformed.Interestinglywiththeset{2,5,47,89,631},alloftheelementsbelongingtoitareprime.

Howmanydistinctsetscontainingeachofthedigitsonethroughnineexactlyoncecontainonlyprimeelements?

Problem119:Digitpowersum

Thenumber512isinterestingbecauseitisequaltothesumofitsdigitsraisedtosomepower:5+1+2=8,and83=512.Anotherexampleofanumberwiththispropertyis614656=284.

Weshalldefineantobethenthtermofthissequenceandinsistthatanumbermustcontainatleasttwodigitstohaveasum.

Youaregiventhata2=512anda10=614656.

Finda30.

Problem120:Squareremainders

Letrbetheremainderwhen(a1)n+(a+1)nisdividedbya2.

Forexample,ifa=7andn=3,thenr=42:63+83=72842mod49.Andasnvaries,sotoowillr,butfora=7itturnsoutthatrmax=42.

For3a1000,findrmax.

Problem121:Discgameprizefund



Abagcontainsonereddiscandonebluedisc.Inagameofchanceaplayertakesadiscatrandomanditscolourisnoted.Aftereachturnthediscisreturnedtothebag,anextrareddiscisadded,andanotherdiscistakenatrandom.

Theplayerpays1toplayandwinsiftheyhavetakenmorebluediscsthanreddiscsattheendofthegame.

Ifthegameisplayedforfourturns,theprobabilityofaplayerwinningisexactly11/120,andsothemaximumprizefundthebankershouldallocateforwinninginthisgamewouldbe10beforetheywouldexpecttoincuraloss.Notethatanypayoutwillbeawholenumberofpoundsandalsoincludestheoriginal1paidtoplaythegame,sointheexamplegiventheplayeractuallywins9.

Findthemaximumprizefundthatshouldbeallocatedtoasinglegameinwhichfifteenturnsareplayed.

Problem122:Efficientexponentiation

Themostnaivewayofcomputingn15requiresfourteenmultiplications:

nn...n=n15

Butusinga"binary"methodyoucancomputeitinsixmultiplications:

nn=n2

n2n2=n4

n4n4=n8

n8n4=n12

n12n2=n14

n14n=n15

Howeveritisyetpossibletocomputeitinonlyfivemultiplications:

nn=n2

n2n=n3

n3n3=n6

n6n6=n12

n12n3=n15

Weshalldefinem(k)tobetheminimumnumberofmultiplicationstocomputenkforexamplem(15)=5.

For1k200,findm(k).

Problem123:Primesquareremainders

Letpnbethenthprime:2,3,5,7,11,...,andletrbetheremainderwhen(pn1)n+(pn+1)nisdividedbypn2.

Forexample,whenn=3,p3=5,and43+63=2805mod25.

Theleastvalueofnforwhichtheremainderfirstexceeds109is7037.

Findtheleastvalueofnforwhichtheremainderfirstexceeds1010.

Problem124:Orderedradicals

Theradicalofn,rad(n),istheproductofthedistinctprimefactorsofn.Forexample,504=23327,sorad(504)=237=42.

Ifwecalculaterad(n)for1n10,thensortthemonrad(n),andsortingonniftheradicalvaluesareequal,weget:

Unsorted Sorted

n rad(n) n rad(n) k1 1 1 1 12 2 2 2 23 3 4 2 3



4 2 8 2 45 5 3 3 56 6 9 3 67 7 5 5 78 2 6 6 89 3 7 7 910 10 10 10 10

LetE(k)bethekthelementinthesortedncolumnforexample,E(4)=8andE(6)=9.

Ifrad(n)issortedfor1n100000,findE(10000).

Problem125:Palindromicsums

Thepalindromicnumber595isinterestingbecauseitcanbewrittenasthesumofconsecutivesquares:62

+72+82+92+102+112+122.

Thereareexactlyelevenpalindromesbelowonethousandthatcanbewrittenasconsecutivesquaresums,andthesumofthesepalindromesis4164.Notethat1=02+12hasnotbeenincludedasthisproblemisconcernedwiththesquaresofpositiveintegers.

Findthesumofallthenumberslessthan108thatarebothpalindromicandcanbewrittenasthesumofconsecutivesquares.

Problem126:Cuboidlayers

Theminimumnumberofcubestocovereveryvisiblefaceonacuboidmeasuring3x2x1istwentytwo.

Ifwethenaddasecondlayertothissoliditwouldrequirefortysixcubestocovereveryvisibleface,thethirdlayerwouldrequireseventyeightcubes,andthefourthlayerwouldrequireonehundredandeighteencubestocovereveryvisibleface.

However,thefirstlayeronacuboidmeasuring5x1x1alsorequirestwentytwocubessimilarlythefirstlayeroncuboidsmeasuring5x3x1,7x2x1,and11x1x1allcontainfortysixcubes.

WeshalldefineC(n)torepresentthenumberofcuboidsthatcontainncubesinoneofitslayers.SoC(22)=2,C(46)=4,C(78)=5,andC(118)=8.

Itturnsoutthat154istheleastvalueofnforwhichC(n)=10.

FindtheleastvalueofnforwhichC(n)=1000.

Problem127:abchits

Theradicalofn,rad(n),istheproductofdistinctprimefactorsofn.Forexample,504=23327,sorad(504)=237=42.

Weshalldefinethetripletofpositiveintegers(a,b,c)tobeanabchitif:

1. GCD(a,b)=GCD(a,c)=GCD(b,c)=12. a



1. GCD(5,27)=GCD(5,32)=GCD(27,32)=12. 5



GiventhatnisapositiveintegerandGCD(n,10)=1,itcanbeshownthattherealwaysexistsavalue,k,forwhichR(k)isdivisiblebyn,andletA(n)betheleastsuchvalueofkforexample,A(7)=6andA(41)=5.

Youaregiventhatforallprimes,p>5,thatp1isdivisiblebyA(p).Forexample,whenp=41,A(41)=5,and40isdivisibleby5.

However,therearerarecompositevaluesforwhichthisisalsotruethefirstfiveexamplesbeing91,259,451,481,and703.

FindthesumofthefirsttwentyfivecompositevaluesofnforwhichGCD(n,10)=1andn1isdivisiblebyA(n).

Problem131:Primecubepartnership

Therearesomeprimevalues,p,forwhichthereexistsapositiveinteger,n,suchthattheexpressionn3+n2pisaperfectcube.

Forexample,whenp=19,83+8219=123.

Whatisperhapsmostsurprisingisthatforeachprimewiththispropertythevalueofnisunique,andthereareonlyfoursuchprimesbelowonehundred.

Howmanyprimesbelowonemillionhavethisremarkableproperty?

Problem132:Largerepunitfactors

Anumberconsistingentirelyofonesiscalledarepunit.WeshalldefineR(k)tobearepunitoflengthk.

Forexample,R(10)=1111111111=11412719091,andthesumoftheseprimefactorsis9414.

FindthesumofthefirstfortyprimefactorsofR(109).

Problem133:Repunitnonfactors

Anumberconsistingentirelyofonesiscalledarepunit.WeshalldefineR(k)tobearepunitoflengthkforexample,R(6)=111111.

LetusconsiderrepunitsoftheformR(10n).

AlthoughR(10),R(100),orR(1000)arenotdivisibleby17,R(10000)isdivisibleby17.YetthereisnovalueofnforwhichR(10n)willdivideby19.Infact,itisremarkablethat11,17,41,and73aretheonlyfourprimesbelowonehundredthatcanbeafactorofR(10n).

FindthesumofalltheprimesbelowonehundredthousandthatwillneverbeafactorofR(10n).

Problem134:Primepairconnection

Considertheconsecutiveprimesp1=19andp2=23.Itcanbeverifiedthat1219isthesmallestnumbersuchthatthelastdigitsareformedbyp1whilstalsobeingdivisiblebyp2.

Infact,withtheexceptionofp1=3andp2=5,foreverypairofconsecutiveprimes,p2>p1,thereexistvaluesofnforwhichthelastdigitsareformedbyp1andnisdivisiblebyp2.LetSbethesmallestofthesevaluesofn.

FindSforeverypairofconsecutiveprimeswith5p11000000.

Problem135:Samedifferences

Giventhepositiveintegers,x,y,andz,areconsecutivetermsofanarithmeticprogression,theleastvalueofthepositiveinteger,n,forwhichtheequation,x2y2z2=n,hasexactlytwosolutionsisn=27:

342272202=1229262=27



Itturnsoutthatn=1155istheleastvaluewhichhasexactlytensolutions.

Howmanyvaluesofnlessthanonemillionhaveexactlytendistinctsolutions?

Problem136:Singletondifference

Thepositiveintegers,x,y,andz,areconsecutivetermsofanarithmeticprogression.Giventhatnisapositiveinteger,theequation,x2y2z2=n,hasexactlyonesolutionwhenn=20:

13210272=20

Infacttherearetwentyfivevaluesofnbelowonehundredforwhichtheequationhasauniquesolution.

Howmanyvaluesofnlessthanfiftymillionhaveexactlyonesolution?

Problem137:Fibonaccigoldennuggets

ConsidertheinfinitepolynomialseriesAF(x)=xF1+x2F2+x3F3+...,whereFkisthekthtermintheFibonaccisequence:1,1,2,3,5,8,...thatis,Fk=Fk1+Fk2,F1=1andF2=1.

ForthisproblemweshallbeinterestedinvaluesofxforwhichAF(x)isapositiveinteger.

SurprisinglyAF(1/2)=(1/2).1+(1/2)2.1+(1/2)3.2+(1/2)4.3+(1/2)5.5+... =1/2+1/4+2/8+3/16+5/32+... =2

Thecorrespondingvaluesofxforthefirstfivenaturalnumbersareshownbelow.

x AF(x)21 11/2 2

(132)/3 3(895)/8 4(343)/5 5

WeshallcallAF(x)agoldennuggetifxisrational,becausetheybecomeincreasinglyrarerforexample,the10thgoldennuggetis74049690.

Findthe15thgoldennugget.

Problem138:Specialisoscelestriangles

Considertheisoscelestrianglewithbaselength,b=16,andlegs,L=17.

ByusingthePythagoreantheoremitcanbeseenthattheheightofthetriangle,h=(17282)=15,whichisonelessthanthebaselength.

Withb=272andL=305,wegeth=273,whichisonemorethanthebaselength,andthisisthesecondsmallestisoscelestrianglewiththepropertythath=b1.

FindLforthetwelvesmallestisoscelestrianglesforwhichh=b1andb,Larepositiveintegers.



Problem139:Pythagoreantiles

Let(a,b,c)representthethreesidesofarightangletrianglewithintegrallengthsides.Itispossibletoplacefoursuchtrianglestogethertoformasquarewithlengthc.

Forexample,(3,4,5)trianglescanbeplacedtogethertoforma5by5squarewitha1by1holeinthemiddleanditcanbeseenthatthe5by5squarecanbetiledwithtwentyfive1by1squares.

However,if(5,12,13)triangleswereusedthentheholewouldmeasure7by7andthesecouldnotbeusedtotilethe13by13square.

Giventhattheperimeteroftherighttriangleislessthanonehundredmillion,howmanyPythagoreantriangleswouldallowsuchatilingtotakeplace?

Problem140:ModifiedFibonaccigoldennuggets

ConsidertheinfinitepolynomialseriesAG(x)=xG1+x2G2+x3G3+...,whereGkisthekthtermofthesecondorderrecurrencerelationGk=Gk1+Gk2,G1=1andG2=4thatis,1,4,5,9,14,23,....

ForthisproblemweshallbeconcernedwithvaluesofxforwhichAG(x)isapositiveinteger.

Thecorrespondingvaluesofxforthefirstfivenaturalnumbersareshownbelow.

x AG(x)(51)/4 1

2/5 2(222)/6 3

(1375)/14 41/2 5

WeshallcallAG(x)agoldennuggetifxisrational,becausetheybecomeincreasinglyrarerforexample,the20thgoldennuggetis211345365.

Findthesumofthefirstthirtygoldennuggets.

Problem141:Investigatingprogressivenumbers,n,whicharealsosquare

Apositiveinteger,n,isdividedbydandthequotientandremainderareqandrrespectively.Inadditiond,q,andrareconsecutivepositiveintegertermsinageometricsequence,butnotnecessarilyinthatorder.

Forexample,58dividedby6hasquotient9andremainder4.Itcanalsobeseenthat4,6,9areconsecutivetermsinageometricsequence(commonratio3/2).Wewillcallsuchnumbers,n,progressive.

Someprogressivenumbers,suchas9and10404=1022,happentoalsobeperfectsquares.Thesumofallprogressiveperfectsquaresbelowonehundredthousandis124657.

Findthesumofallprogressiveperfectsquaresbelowonetrillion(1012).

Problem142:PerfectSquareCollection

Findthesmallestx+y+zwithintegersx>y>z>0suchthatx+y,xy,x+z,xz,y+z,yzareallperfectsquares.

Problem143:InvestigatingtheTorricellipointofatriangle



LetABCbeatrianglewithallinterioranglesbeinglessthan120degrees.LetXbeanypointinsidethetriangleandletXA=p,XC=q,andXB=r.

FermatchallengedTorricellitofindthepositionofXsuchthatp+q+rwasminimised.

TorricelliwasabletoprovethatifequilateraltrianglesAOB,BNCandAMCareconstructedoneachsideoftriangleABC,thecircumscribedcirclesofAOB,BNC,andAMCwillintersectatasinglepoint,T,insidethetriangle.MoreoverheprovedthatT,calledtheTorricelli/Fermatpoint,minimisesp+q+r.Evenmoreremarkable,itcanbeshownthatwhenthesumisminimised,AN=BM=CO=p+q+randthatAN,BMandCOalsointersectatT.

Ifthesumisminimisedanda,b,c,p,qandrareallpositiveintegersweshallcalltriangleABCaTorricellitriangle.Forexample,a=399,b=455,c=511isanexampleofaTorricellitriangle,withp+q+r=784.

Findthesumofalldistinctvaluesofp+q+r120000forTorricellitriangles.

Problem144:Investigatingmultiplereflectionsofalaserbeam

Inlaserphysics,a"whitecell"isamirrorsystemthatactsasadelaylineforthelaserbeam.Thebeamentersthecell,bouncesaroundonthemirrors,andeventuallyworksitswaybackout.

Thespecificwhitecellwewillbeconsideringisanellipsewiththeequation4x2+y2=100

Thesectioncorrespondingto0.01x+0.01atthetopismissing,allowingthelighttoenterandexitthroughthehole.



Thelightbeaminthisproblemstartsatthepoint(0.0,10.1)justoutsidethewhitecell,andthebeamfirstimpactsthemirrorat(1.4,9.6).

Eachtimethelaserbeamhitsthesurfaceoftheellipse,itfollowstheusuallawofreflection"angleofincidenceequalsangleofreflection."Thatis,boththeincidentandreflectedbeamsmakethesameanglewiththenormallineatthepointofincidence.

Inthefigureontheleft,theredlineshowsthefirsttwopointsofcontactbetweenthelaserbeamandthewallofthewhitecellthebluelineshowsthelinetangenttotheellipseatthepointofincidenceofthefirstbounce.

Theslopemofthetangentlineatanypoint(x,y)ofthegivenellipseis:m=4x/y

Thenormallineisperpendiculartothistangentlineatthepointofincidence.

Theanimationontherightshowsthefirst10reflectionsofthebeam.

Howmanytimesdoesthebeamhittheinternalsurfaceofthewhitecellbeforeexiting?

Problem145:Howmanyreversiblenumbersaretherebelowonebillion?

Somepositiveintegersnhavethepropertythatthesum[n+reverse(n)]consistsentirelyofodd(decimal)digits.Forinstance,36+63=99and409+904=1313.Wewillcallsuchnumbersreversibleso36,63,409,and904arereversible.Leadingzeroesarenotallowedineithernorreverse(n).

Thereare120reversiblenumbersbelowonethousand.

Howmanyreversiblenumbersaretherebelowonebillion(109)?

Problem146:InvestigatingaPrimePattern

Thesmallestpositiveintegernforwhichthenumbersn2+1,n2+3,n2+7,n2+9,n2+13,andn2+27areconsecutiveprimesis10.Thesumofallsuchintegersnbelowonemillionis1242490.

Whatisthesumofallsuchintegersnbelow150million?

Problem147:Rectanglesincrosshatchedgrids

Ina3x2crosshatchedgrid,atotalof37differentrectanglescouldbesituatedwithinthatgridasindicatedinthesketch.

Thereare5gridssmallerthan3x2,verticalandhorizontaldimensionsbeingimportant,i.e.1x1,2x1,3x1,1x2and2x2.Ifeachofthemiscrosshatched,thefollowingnumberofdifferentrectanglescouldbesituatedwithinthosesmallergrids:

1x1:12x1:43x1:81x2:42x2:18

Addingthosetothe37ofthe3x2grid,atotalof72differentrectanglescouldbesituatedwithin3x2andsmallergrids.

Howmanydifferentrectanglescouldbesituatedwithin47x43andsmallergrids?

Problem148:ExploringPascal'striangle



WecaneasilyverifythatnoneoftheentriesinthefirstsevenrowsofPascal'strianglearedivisibleby7:

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 11 6 15 20 15 6 1

However,ifwecheckthefirstonehundredrows,wewillfindthatonly2361ofthe5050entriesarenotdivisibleby7.

Findthenumberofentrieswhicharenotdivisibleby7inthefirstonebillion(109)rowsofPascal'striangle.

Problem149:Searchingforamaximumsumsubsequence

Lookingatthetablebelow,itiseasytoverifythatthemaximumpossiblesumofadjacentnumbersinanydirection(horizontal,vertical,diagonalorantidiagonal)is16(=8+7+1).

2 5 3 2

9 6 5 1

3 2 7 3

1 8 4 8

Now,letusrepeatthesearch,butonamuchlargerscale:

First,generatefourmillionpseudorandomnumbersusingaspecificformofwhatisknownasa"LaggedFibonacciGenerator":

For1k55,sk=[100003200003k+300007k3](modulo1000000)500000.For56k4000000,sk=[sk24+sk55+1000000](modulo1000000)500000.

Thus,s10=393027ands100=86613.

Thetermsofsarethenarrangedina20002000table,usingthefirst2000numberstofillthefirstrow(sequentially),thenext2000numberstofillthesecondrow,andsoon.

Finally,findthegreatestsumof(anynumberof)adjacententriesinanydirection(horizontal,vertical,diagonalorantidiagonal).

Problem150:Searchingatriangulararrayforasubtrianglehavingminimumsum

Inatriangulararrayofpositiveandnegativeintegers,wewishtofindasubtrianglesuchthatthesumofthenumbersitcontainsisthesmallestpossible.

Intheexamplebelow,itcanbeeasilyverifiedthatthemarkedtrianglesatisfiesthisconditionhavingasumof42.

Wewishtomakesuchatriangulararraywithonethousandrows,sowegenerate500500pseudorandomnumbersskintherange219,usingatypeofrandomnumbergenerator(knownasaLinearCongruentialGenerator)asfollows:

t:=0fork=1uptok=500500:t:=(615949*t+797807)modulo220



sk:=t219

Thus:s1=273519,s2=153582,s3=450905etc

Ourtriangulararrayisthenformedusingthepseudorandomnumbersthus:

s1s2s3

s4s5s6s7s8s9s10

...

Subtrianglescanstartatanyelementofthearrayandextenddownasfaraswelike(takinginthetwoelementsdirectlybelowitfromthenextrow,thethreeelementsdirectlybelowfromtherowafterthat,andsoon).The"sumofasubtriangle"isdefinedasthesumofalltheelementsitcontains.Findthesmallestpossiblesubtrianglesum.

Problem151:Papersheetsofstandardsizes:anexpectedvalueproblem

Aprintingshopruns16batches(jobs)everyweekandeachbatchrequiresasheetofspecialcolourproofingpaperofsizeA5.

EveryMondaymorning,theforemanopensanewenvelope,containingalargesheetofthespecialpaperwithsizeA1.

Heproceedstocutitinhalf,thusgettingtwosheetsofsizeA2.ThenhecutsoneoftheminhalftogettwosheetsofsizeA3andsoonuntilheobtainstheA5sizesheetneededforthefirstbatchoftheweek.

Alltheunusedsheetsareplacedbackintheenvelope.

Atthebeginningofeachsubsequentbatch,hetakesfromtheenvelopeonesheetofpaperatrandom.IfitisofsizeA5,heusesit.Ifitislarger,herepeatsthe'cutinhalf'procedureuntilhehaswhatheneedsandanyremainingsheetsarealwaysplacedbackintheenvelope.

Excludingthefirstandlastbatchoftheweek,findtheexpectednumberoftimes(duringeachweek)thattheforemanfindsasinglesheetofpaperintheenvelope.

Giveyouranswerroundedtosixdecimalplacesusingtheformatx.xxxxxx.

Problem152:Writing1/2asasumofinversesquares

Thereareseveralwaystowritethenumber1/2asasumofinversesquaresusingdistinctintegers.

Forinstance,thenumbers{2,3,4,5,7,12,15,20,28,35}canbeused:

Infact,onlyusingintegersbetween2and45inclusive,thereareexactlythreewaystodoit,theremainingtwobeing:{2,3,4,6,7,9,10,20,28,35,36,45}and{2,3,4,6,7,9,12,15,28,30,35,36,45}.

Howmanywaysaretheretowritethenumber1/2asasumofinversesquaresusingdistinctintegersbetween2and80inclusive?

Problem153:InvestigatingGaussianIntegers



Asweallknowtheequationx2=1hasnosolutionsforrealx.Ifwehoweverintroducetheimaginarynumberithisequationhastwosolutions:x=iandx=i.Ifwegoastepfurthertheequation(x3)2=4hastwocomplexsolutions:x=3+2iandx=32i.x=3+2iandx=32iarecalledeachothers'complexconjugate.Numbersoftheforma+biarecalledcomplexnumbers.Ingenerala+biandabiareeachother'scomplexconjugate.

AGaussianIntegerisacomplexnumbera+bisuchthatbothaandbareintegers.TheregularintegersarealsoGaussianintegers(withb=0).TodistinguishthemfromGaussianintegerswithb0wecallsuchintegers"rationalintegers."AGaussianintegeriscalledadivisorofarationalintegerniftheresultisalsoaGaussianinteger.Ifforexamplewedivide5by1+2iwecansimplify inthefollowingmanner:

Multiplynumeratoranddenominatorbythecomplexconjugateof1+2i:12i.

Theresultis .

So1+2iisadivisorof5.Notethat1+iisnotadivisorof5because .

NotealsothatiftheGaussianInteger(a+bi)isadivisorofarationalintegern,thenitscomplexconjugate(abi)isalsoadivisorofn.

Infact,5hassixdivisorssuchthattherealpartispositive:{1,1+2i,12i,2+i,2i,5}.Thefollowingisatableofallofthedivisorsforthefirstfivepositiverationalintegers:

n GaussianintegerdivisorswithpositiverealpartSums(n)ofthesedivisors

1 1 12 1,1+i,1i,2 53 1,3 44 1,1+i,1i,2,2+2i,22i,4 135 1,1+2i,12i,2+i,2i,5 12

Fordivisorswithpositiverealparts,then,wehave: .

For1n105,s(n)=17924657155.

Whatiss(n)for1n108?

Problem154:ExploringPascal'spyramid

Atriangularpyramidisconstructedusingsphericalballssothateachballrestsonexactlythreeballsofthenextlowerlevel.

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