Re reational Mathemati sColloquium IIBOOK OF ABSTRACTS

University of ÉvoraApril 27th - April 30th, 2011

Organization:Ludus Asso iationUniversity of ÉvoraMuseum of S ien e, University of LisbonOrganizing Committee:Alda Carvalho (ISEL, Portugal)Ana Santos (University of Évora, Portugal)Carlos Santos (ISEC, Portugal)Jorge Nuno Silva (University of Lisbon, Portugal)Liliana Monteiro (University of Lisbon, Portugal)Sandra Vinagre (University of Évora, Portugal)S ienti� Committee:David Singmaster (London South Bank University, England)João Pedro Neto (University of Lisbon, Portugal)Jorge Bues u (University of Lisbon, Portugal)Jorge Nuno Silva (University of Lisbon, Portugal)Keith Devlin (Stanford University, England)Nuno Crato (Te hni al University of Lisbon, Portugal)Ri hard Nowakowski (Dalhousie University, Canada)Robin Wilson (Open University, England)Sandra Vinagre (University of Évora, Portugal)Sponsors:

Contents1 Foreword 32 Invited Speakers 53 Program 94 Abstra ts 135 Mailing List 25

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1 Foreword�Re reational Mathemati s� is a problemati expression. For some people,like most professional mathemati ians, Mathemati s is lots of fun; but forothers, like some students, Mathemati s an be a nightmare.Histori ally, we know that some mathemati al resear h areas are deeplylinked to puzzles and games, probability and han e games, graph theoryand the Brigdes of Königsberg.The University of Évora, the Ludus Asso iation, and the Museum of S ien eof the University of Lisbon organize the Re reational Mathemati s Collo-quium II.Our Colloquium will be a Show and Tell of bright pearls of Mathemati s,with varied levels of sophisti ation, entertaining many audien es. Its maingoal is to foster mathemati al appre iation, an important step if we are tosee improvements in its pra ti e. More information an be onsulted in thefollowing web page:http://ludi um.org/rm11/ The Organizing Committee

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2 Invited SpeakersColin Wright(born in Australia, President of the Liverpool Mathemati al So iety)

Colin Wright graduated in 1982 from Monash University, Melbourne, Aus-tralia, with a B.S .(Hons) in Pure Mathemati s, and went on to re eivehis do torate in 1990 from Cambridge University, England. While at Cam-bridge he also learned how to �re breathe, uni y le, juggle and ballroomdan e. Sin e then he has worked as a resear h mathemati ian, a omputerprogrammer, and an ele troni s hardware designer. Colin also enjoys sail-ing small boats and playing bridge, although not (so far) at the same time.Colin is a o-founder of Solipsys Limited along with Andrew Lipson and heis the presenter of the popular s ien e Juggling Talk Series. He is a formerPresident of the Liverpool Mathemati al So iety.Edward Pegg(born in USA, Wolfram Resear h)

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Edward Pegg is an expert on mathemati al puzzles and is a self-des ribedre reational mathemati ian. He reates puzzles for the Mathemati al Asso- iation of Ameri a online at Ed Pegg, Jr.'s Math Games. His puzzles havealso been used by Will Shortz on the puzzle segment of NPR's Weekend Edi-tion Sunday. In 2000, he left NORAD to join Wolfram Resear h, where he ollaborated on A New Kind of S ien e (NKS). In 2004 he started assistingEri W. Weisstein at Wolfram MathWorld. He has made ontributions toseveral hundred MathWorld arti les. He is one of the hief onsultants forNumb3rs.Keith Devlin(born in England, Stanford University)Dr. Keith Devlin is a o-founder and Exe utive Dire tor of the Stanford Uni-versity's H STAR institute, a o-founder of the Stanford Media X resear hnetwork, and a Senior Resear her at CSLI. He is a World E onomi ForumFellow and a Fellow of the Ameri an Asso iation for the Advan ement ofS ien e. His urrent resear h is fo used on the use of di�erent media totea h and ommuni ate mathemati s to diverse audien es. He also works onthe design of information/reasoning systems for intelligen e analysis. Otherresear h interests in lude: models of reasoning, appli ations of mathemati- al te hniques in the study of ommuni ation, and mathemati al ognition.He has written 28 books and over 80 published resear h arti les. Re ipi-ent of the Pythagoras Prize, the Peano Prize, the Galileo Prize, the CarlSagan Award, and the Joint Poli y Board for Mathemati s Communi ationsAward. In 2003, he was re ognized by the California State Assembly for his�innovative work and longtime servi e in the �eld of mathemati s and itsrelation to logi and linguisti s.� He is �the Math Guy� on National Publi Radio.We hardly suggest the site www.stanford.edu/ ∼kdevlin.

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Lennart Green(born in Sweden, World Champion lose-up/ ard Magi ian)

Lennart Green is known worldwide as the E entri Swedish Misdire tor. Heis a master of mental manipulation, misdire tion and magi entertainment!He is a spe ialist in rooked gambling, rigged bets, puzzles and paradoxes: AWorld Champion in Card Magi . Lennart invented his te hnique by observ-ing inmates and patients in prison whilst studying medi ine and psy hology.After 15 years of pra ti e as a General Pra titioner, he de ided to pursue anew areer fo used on his love for magi and entertainment. He says he wasinspired by the reative methods of some on artists!Lennart travels the world giving motivational talks, le tures, publi per-forman es (e.g., TED in Monterey) and TV shows. His interests/hobbiesin lude repairing old violins, olle ting strange obje ts, enigma-puzzles, op-ti al illusions and spe ial books. He enjoys legends, proverbs, aphorisms,and mottos like:It doesn't matter how fast you run - If you run in the wrong dire tion.Ri hard Nowakowski(born in England, Dalhousie University)

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Ri hard Nowakowski is a Professor at Dalhousie University in Canada. Hisresear h interests in lude graph theory and ombinatorial game theory buteven his graph theory work involves games and puzzles su h as ops-and-robbers and leaning a graph. He has helped organize the Games-of-no-Chan e onferen es at MSRI and BIRS and has edited the pro eedings.Sara Santos(born in Portugal, Royal Institution of Great Britain)

Sara Santos is Clothworkers' Fellow for Mathemati s at the Ri, responsiblefor the Mathemati s Master lasses for se ondary s hool students. Sara isone of the founders of Maths Busking, award winning proje t for Seed ofS ien e 2011 on the ategory of s ien e ommuni ation. Sara's desert islandbook is Byrne's Six Books of Eu lid.

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3 ProgramWednesday, 27th April8:30 Wel ome and registration9:15 Opening Session9:30 Re ently Solved ProblemsEdward Pegg, Wolfram Resear h10:30 Bodies Invisible in Several Dire tionsVera Rosh hina, CIMA-UE, University of Évora11:00 Co�ee-Break11:30 Fooling around with Stable MarriagesJorge Bues u, University of Lisbon12:00 Syzygies played by elementary s hool studentsDores Ferreira, University of Minho12:30 Vogeler's Diagram and Rational Derived PolynomialsAlda Carvalho, ISEL13:00 Break for lun h14:30 Juggling: Theory and Pra ti eColin Wright, Liverpool Mathemati al So iety15:30 Medieval ProblemsJoaquim Euri o Nogueira, New University of Lisbon16:00 Co�ee-Break16:30 The Football Pool ProblemAntónio Ma hiavelo, University of Oporto17:00 Shannon Swit hing Game and VariantsIlda Perez, University of Lisbon9

17:30 A dynami al approa h to ne kla es and wordsCristina Serpa, University of Lisbon18:00 Mathemati al Cir usAdelaide Carreira, University of Lisbon19:00 Wel ome Co ktail (Évora Town Hall)Thursday, 28th April9:30 Master lasses and Maths BuskingSara Santos, Royal Institution of Great Britain10:30 On a Certain Chaoti GameCarlos Ramos, University of Évora11:00 Co�ee-Break11:30 Using Empiri al Map Proje tions for Resear h, Edu ation and FunJoaquim Alves Gaspar, University of Lisbon12:00 Some Math Problems with Trains and RailwaysHelder Pinto, University of Lisbon12:30 Predominan e GameHelena Sousa Melo, University of Azores13:00 Break for lun h14:30 Visit to Vila Viçosa

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Friday, 29th April9:30 Games for Lazy but Intelligent ParentsRi hard Nowakowski, Dalhousie University10:30 The shape of the sound: from bird singing to western musi Carlota Simões, University of Coimbra11:00 Co�ee-Break11:30 The Mathemati s of Cartoons: A Brief SurveyNatália Bebiano, University of Coimbra12:00 Extreme Alphameti sMi hael Keith12:30 SangakuAntonieta Constantino, ES de Mira�ores13:00 Break for lun h14:30 Magi ShowLennart Green, World Champion lose-up/ ard Magi ian15:30 Mathemati al QuiltsAndreia Hall, University of Aveiro16:00 Co�ee-Break16:30 Bo age and Mathemati sFilipe Papança, Militar A ademy17:00 Five Years of Ludus Asso iationCarlos Pereira dos Santos, ISEC17:30 Leonardo of Pisa (Fibona i)Keith Devlin, Stanford University18:30 Wine Tasting (Rota dos Vinhos do Alentejo)19:30 Conferen e Dinner11

Saturday, 30th April9:30 What if all middle s hool mathemati s edu ationwere presented as re reational mathemati s?Keith Devlin, Stanford University10:30 One Noti es on the Creation and use ofPedagogi al Games in the Mathemati s LessonsAna Júlia Martins, University of Évora and ISEC11:00 Co�ee-Break11:30 Winning Nim with Beatty and Fibona iMaria Joana Torres, University of Minho12:00 Treason GameJoão Cabral, University of Azores12:30 Slimetrail for the Visual ImpairedCarlota Dias, High S hool Matias Aires13:00 Break for lun h14:30 The Last Geometri Argument in the History of PiCarlos Pereira dos Santos, ISEC15:30 Games ClubeMathMaria do Céu Soares, New University of Lisbon16:00 Closing Session

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4 Abstra tsMathemati al Cir usAdelaide Carreira, University of LisbonAlexandre Silva, University of LisbonAna Eliete, University of LisbonAnabela Teixeira, Museum of S ien e, University of LisbonAndreia Hall, University of AveiroFilomena Carreira, University of LisbonInês Leitao, University of LisbonJoana Fernandes, University of LisbonJorge Nuno Silva, University of LisbonMarta Duarte, University of LisbonTeresa Iná io, University of LisbonTiago Santos, University of LisbonValter Nunes, University of LisbonInspiring, informative, entertaining and astonishing. This is our ir us:Good mathemati s, lots of fun!

Five Years of Ludus Asso iationAlda Carvalho, ISELAntonieta Constantino, ES Mira�oresCarlos Pereira dos Santos, ISECCarlota Dias, High S hool Matias AiresHelder Pinto, University of LisbonJoão Pedro Neto, University of LisbonJorge Nuno Silva, University of LisbonLiliana Monteiro, University of LisbonSandra Vinagre, University of ÉvoraLudus is 5 years old. Portugal is not the same sin e the hild was born!13

Vogeler's Diagram and Rational Derived PolynomialsAlda Carvalho, ISELCarlos Pereira dos Santos, ISECStudents usually don't like problems with irrational solutions: humans likethe whole numbers! In this work we present some ideas on the onstru tionof problems (geometry and al ulus) with whole solutions. We illustrate thelittle-known Vogeler's diagram and the problem of rational derived polyno-mials.Hidden Se rets of Times-TablesAndreia Hall, University of AveiroRosa Amélia Martins, University of AveiroMathemati s play an important role in quilt making. Tiling, symmetry,fra tals, reptiles and Voronoi diagrams are just a few of many mathemati al on epts that an be used and explored in pat hwork. In this talk we shallpresent some quilt examples that use mathemati al models.One Noti es on the Creation and use ofPedagogi al Games in the Mathemati s LessonsAna Júlia Martins, University of Évora and ISECWith this resear h we intend to ontribute for the improvement of mathemat-i s' learning results of the students of third and fourth grades of ElementaryS hool of Serra da Vila . In this way, we tried to reate pedagogi al games toindu e in pupils the motivation for learning this subje t. With this purpose,we reated and used games based on the preferen es that students revealedin s hool and in the di� ulties they have showed in the appli ation of some urri ular on epts.It is an almost-experimental study, where two lasses from the third andfourth grades of Elementary s hools parti ipated. The lass from Serra daVila s hool worked as an experimental group and the lass of Bordinheiras hool as a ontrol one.In the �nal part of the work, we ompare the results obtained by the studentsbefore and after the appli ation of the games, and we found that after its usethere was a positive hange in their results. Thus, we on luded that thesegames reated from tastes and preferen es of pupils show some potential onimproving pupils' attitude in the lass under study towards math lessons,and also in obtaining better results in the evaluation tests, allowing to in- rease students' su ess in this area. 14

SangakuAntonieta Constantino, ES de Mira�oresSangaku are Japanese geometri al puzzles painted on wooden tablets reatedduring the Edo period (1603-1867), when Japan was isolated from the restof the world. The Sangaku hung in the pre in ts of shrines as o�erings tothe gods.

The Football Pool ProblemAntónio Ma hiavelo, University of OportoFootball pools gave rise to a ombinatorial problem related to oding or-re ting errors that is still far from solved. In this talk we will des ribe thisproblem, some of what is known about it, and some of what we would liketo know. But mostly, we will try to show how to have some real fun with it!The Last Geometri Argument in the History of πCarlos Pereira dos Santos, ISECAlda Carvalho, ISELAr himedes presented a lassi al geometri method to obtain a good ap-proximation of π. Other issues, su h as a proof of the irrationality of π wereonly possible with the rising of al ulus (it is still not known any geometri proof). In this work we present a very re ent geometri justi� ation relatedto π (presented in a Wastlund's paper (Am. Monthly, 2007)) and its onse-quen es.15

On a Certain Chaoti GameCarlos Ramos, University of ÉvoraMarta Riera, University of ÉvoraWe propose a family of games whi h involves the haoti behavior of a purelyme hani al system. Perturbations in the game parameters hanges the gamefrom a trivial one to an almost purely han e game. We dis uss if, in betweenthe two extreme ases, we an (or not) obtain interesting games, involvingability, per eption and strategy.Slimetrail for the Visual ImpairedCarlota Dias, High S hool Matias AiresPedro Palhares, Institute of Edu ation, University of MinhoJorge Nuno Silva, University of LisbonOur per eption of the world around us depends on our visual per eption,the same happens in identifying and learning a mathemati al game. The hild with poor vision has other di� ulties when apprehending a game, andits adaptations must take dimension and olor of the material into a ount.�The �nal area in whi h information must be olle ted is the a essibilityof both materials and the presentation of information. (...)� (Castellano,2010a).A ording to Blan o & Rubio (1993) most of our judgments on erning whatgoes on around us, under normal ir umstan es, in luding a signi� ant partof our knowledge of the world and ourselves, present themselves in the formof visual images. Thus, being blind keeps one from a essing a ertain typeof information, whi h makes it impossible to represent the world as beingseen.The reper ussions of the visual de� its a�e t merely adaptive questions,known as A tivities of Daily Life, as well as the apa ity to a ess ulturalinformation through the usual means. Therefore, a visually impaired personmust optimize his/her sensory systems.A blind hild apprehends a game mainly by tou h. �Blind people re ognizeobje ts by tou h or by sound. Re ognition is not dependent on a parti ularsensory modality. (...)� (Smith, 2009). Ta tile identi� ation is essential tolearn the rules of a game to handle its pawns and identify the moves. Thehapti system is the foundation of this whole pro ess.�The hapti system, unlike the other per eptual systems, in ludes the wholebody, most of its parts, and all of its surfa e. The extremities are exploratorysense organs, but they are also perforatory motor organs; that is to say, the16

equipment for feeling is anatomi ally the same as the equipment for doing.This ombination is not found in the o ular or the auditory system. We anexplore things with the eyes but not alter the environment; however we anexplore and alter the environment with hands. (...)� (Gibson, 1966).But playing games is also understood as a method of a quiring so ial skillsas soon as possible. �Get your hild ready for the many stru tured andunstru tured play situations he/she will en ounter in the early years.(...)�(Castellano, 2010b).The prime obje tive of this resear h is the reation of onditions that enablethe parti ipation of hildren and youngsters with visual handi ap in the Na-tional Championship of Mathemati al Games, as well as the implementationof the pra ti e of its games in the edu ational system, developing gamingskill through the reation and perfe tion of strategies. The method usedfalls in the qualitative resear h, with information being olle ted through di-re t observation in two separate stages. During the �rst stage a series of teststo the di�erent game boards and respe tive pawns with variety in dimension,form, texture and olor were ondu ted. A set a rules in braille, enlarged, ororally explained, depending on the needs of the student, was made available.The se ond observation stage assumes that the student already has a gooddomain of the game that allows him or her to develop skills at the levels of ommuni ation and strategy reation. The investigation proje t is, at thistime, at the end of stage one. The evolution pro ess of the adapted gameSlimetrail is being presented in this ommuni ation. It resulted in two dif-ferent versions, one for low vision and another for blindness.Referen es:Blan o, F. & Rubio, M. (1993). Per eption Sin Vison In Psi ología de laCeguera, 3, (p. 51-110). Madrid: Alianza Editorial.Castellano, C. (2010a). A ademi s In Getting Ready for College. Beginsin Third Grade, 2, (p. 17). Charllote, North Carolina: Information AgePublishing, In .Castellano, C. (2010b). So ial Awareness and So ial Skills In Getting Readyfor College. Begins in Third Grade, 5, (p. 71). Charllote, North Carolina:Information Age Publishing, In .Gibson, J. (1966). The Hapti System and its Components In The SensesConsidered as Per eptual System, VI, (p. 99). Westport: Greenwood Press,Publishers.Smith, E. & Kosslyn, S. (2009). A hieving Visual re ognition: Have I seenyou before? In Cognitive Psy hology, 2, (p. 70). New Jersey: Pearson Edu- ation. 17

The shape of the sound: from bird singing to western musi Carlota Simões, University of CoimbraThe solid but dis rete relationship between Mathemati s and Musi is atleast as old as Pythagoras. Mathemati s relates with Musi at the momentof tuning an instrument, sin e onsonant notes are produ ed by sounds whosefrequen ies have interesting mathemati al relationships.The problem of tuning a musi al instrument began as a problem of propor-tions, when the instrument was the vibrating string, but be ame even moreinteresting when the piano was reated, requiring a tempered tuning dis ov-ered only after the intervention of mathemati ians.However, without any knowledge of arithmeti s, birds from all over the worlda urately reprodu e the several intervals of the diatoni s ale and singmelodies in di�erent musi al s ales reated by man throughout the ages,from the Greek modes to the diatoni s ales of our days.Juggling: Theory and Pra ti eColin Wright, Liverpool Mathemati al So ietyJuggling has fas inated people for enturies. Seemingly oblivious to gravity,the skilled pra titioner will keep several obje ts in the air at one time, andweave omplex patterns that seem to defy analysis.In this talk the speaker demonstrates a sele tion of the patterns and skillsof juggling while at the same time developing a simple method of des ribingand annotating a lass of juggling patterns. By using elementary mathemat-i s these patterns an be lassi�ed, leading to a simple way to des ribe thosepatterns that are known already, and a te hnique for dis overing new ones.A dynami al approa h to ne kla es and wordsCristina Serpa, University of LisbonJorge Bues u, University of LisbonWe use on epts from dynami al systems, namely periodi orbits in ir lemaps, together with ombinatorial and ounting arguments, to generate ape-riodi ne kla es and Lyndon words.18

Syzygies played by elementary s hool studentsDores Ferreira, University of MinhoPedro Palhares, University of MinhoJorge Nuno Silva, University of LisbonThe mathemati ian Charles Lutwidge Dodgson, well known by the pseudonymof Lewis Carroll, besides being the famous author of �Ali e's adventures inwonderland� he was also the inventor of a variety of games and puzzles.Lewis Carroll liked to play with words and one of his inventions was a word-puzzle that he named Syzygies. As part of a resear h involving games, inthe last few months elementary s hool students have played Syzygies. Inthis presentation we will present the results of this pra ti e as well as thelearning pro ess and others possible onne tions with mathemati s.Re ently Solved ProblemsEdward Pegg, Wolfram Resear hToday, nearly everyone has a ess to a super omputer. This has allowedmany problems to be expanded or solved, su h as Paterson's Worms, Mrs.Perkins's Quilts, Fusible numbers, Rubik's ube, the �rst 5 trillion digits ofPi, Setups for Texas Holdem Poker, and other problems. There are someproblems that haven't been looked at re ently, su h as the no-3-in-a-lineproblem. The talk will answer �What an you do with a super omputer?�Bo age and Mathemati sFilipe Papança, Militar A ademyThis ommuni ation analyzes the Mathemati knowledge in some poetry ofBo age and the politi and s ienti� ontext in the se ond part of the eigh-teen entury. Also, it analyzes the in�uen e of his military formation ina quisition of Mathemati knowledge.Shannon Swit hing Game and VariantsIlda Perez, University of LisbonThe Shannon swit hing game, introdu ed by R.W. Shannon ir a 1960, is atwo-player game on an undire ted graph with two distinguished verti es. Theplayers, Join and Cut, alternately hoose one unplayed edge of the graph.Join makes the hosen edge invulnerable to deletion and Cut deletes the ho-sen edge. The obje tive of Join is to make invulnerable a path onne tingthe distinguished verti es while the obje tive of Cut is to prevent Join fromsu eeding.The game was ompletely solved by A. Lehman in 1964 by onsidering its19

natural extension to matroids. In 1987, Y.O. Hamidoune and M.Las Vergnasintrodu ed and solved dire ted versions of the game for graphs, onje turingnaturally their generalization to oriented matroids. Despite some re ent re-sults Hamidoune-Las Vergnas onje tures remain wide open. We review themain ideas of the analysis of the graphi al, undire ted and dire ted, games.Leonardo of Pisa (Fibona i)Keith Devlin, Standford UniversityA new book on Leonardo of Pisa (Fibona i) is oming out in July. This bookis the �rst ever omplete biography of him. This talk is about Leonardo's ontribution to establishing modern re reational mathemati s.

What if all middle s hool mathemati s edu ation were presented as re re-ational mathemati s?Keith Devlin, Standford UniversityWith videogames, this is starting to look like a real possibility. To be sure,the term �re reational mathemati s� has ome to be regarded as meaning aparti ular subset of mathemati s, but that is largely a result of the mediaof presentation. At heart, re reational mathemati s is about doing math-emati s as a pleasant pastime � for fun. Videogames o�er an opportunityto present the entire middle s hool mathemati s urri ulum in that fashion.My talk is based on �ve-years experien e working with a large videogame ompany in Sili on Valley. 20

Some Math Problems with Trains and RailwaysHelder Pinto, University of LisbonOne of the most important tea her's tasks is to �nd problems that are ap-pealing to their students. There are many interesting problems/puzzles thatare true lassi s (for example, the riddles of Sam Loyd, Henry Dudeney andLewis Carroll), but do not always have a su� iently appealing and attra tive ontext to today students - although they remain urrent and relevant, intheir mathemati al aspe ts, as the day they were reated. However, witha little imagination and work, these problems an be adapted to di�erent ontexts, more modern as, for example, transports, sports, and the everydaylife. The talk that will be presented will show some examples of well knownproblems/puzzles whi h have been adapted to trains and railways (theme,so the author has found, that seems to be part of the imaginary of everyone,whether kids or grown-ups) and published on a book by the Portuguese So- iety of Mathemati s (Matemáti a & Comboios, 2010). The hoi e of thisparti ular theme is due to the author's own interest in this subje t and willtry to show that the same mathemati al problem/puzzle may have variousformulations in the quest to �nd something that is truly meaningful and in-teresting to the audien e (usually hildren and adoles ents) that is intended.Predominan e GameHelena Sousa Melo, University of AzoresJoão Cabral, University of AzoresPredominan e is a new strategy game for two-player played on a Penrose'stiling board with a time ontrol. This game is played in two stages. The �rststage is played by the two players putting in position the seventy two pie eson board. The pie es are divided in white and bla k sets, pla ed in an alter-nate way. Ea h set has four di�erent ranking pie es, ordered by value: four ommanders, six knights, ten soldiers and sixteen ons ripts. In the se ondstage of the game, the players an move the pie es and make aptures. Themovement without apture is hara terized by a path on the board that isformed by only adja ent empty ells, but a movement with the intention of apture an be made over non-empty ells. The apture is made repla ingone pie e, the aptured, by the other, the aptor. The ommander movesthe maximum of four adja ent ells; the knight moves the maximum of threeadja ent ells; the soldier two ells, and the ons ript one ell. The pie e an only apture pie es of the same ranking or below. Predominan e is a unning game, playable at several levels of omplexity. The game ends whena player rea h forty �ve points in aptured pie es. Some winning strategiesare explored. 21

Treason GameJoão Cabral, University of AzoresHelena Sousa Melo, University of AzoresTreason is a two player game played with the usual rules of the he kersgame, but played in a di�erent board, with the needed adaptations. Insteadof squares we have triangles, and the initial position of the pie es is similarto the he kers game. The board has also in the main diagonal a free battlezone that allows the players to diversify the strategies until a level of om-plexity very high. One of the main di�eren es between this game and the he kers is that a player has, at least, from one triangle, three possibilitylines to play. The name of the game omes from the most important movethat allows one player to promote pawns to queens, behind enemy lines. Thismove gives a han e to queens to sweep literally the enemy pawns by theba k, working as a spy in enemy lines. If we imagine a movement of a realsoldier with the sword in the hand, he an kill to the left, to the right, infront or to the ba k. This game reprodu es exa tly these moves, whi h arelimited in the usual he kers game.Using Empiri al Map Proje tions for Resear h, Edu ation and FunJoaquim Alves Gaspar, University of LisbonA numeri al model based on the on ept of 'multidimensional s aling', heregeneralized to spheri al distan es and dire tions, was developed and tested,with the obje tive of simulating the main geometri features of early nauti al harts. Starting with a sample of routes de�ned on the spheri al surfa e ofthe Earth, the pro ess onsists in re-arranging the positions of the pointson a plane so that the di�eren es between the initial (spheri al) and �nal(planar) distan es and dire tions between them are minimized. The modelproved to be an e�e tive and easy-to-use resear h tool and may be used, notonly for simulating and assessing the various fa tors a�e ting the geometryof early harts, but also for edu ational purposes e.g. for illustrating theproperties of map proje tions.Medieval ProblemsJoaquim Euri o Nogueira, New University of LisbonThe �desert- rossing problems� (also known as the �Jeep problems�) havebeen around sin e Al uin of York presented some earlier versions to Charle-magne, emperor of the Fran s. In this talk I give a brief a ount of thehistory of these problems sin e Al uin's era to modern times, explainingtheir solutions in full detail. 22

Fooling around with Stable MarriagesJorge Bues u, University of LisbonIn this talk we address a lassi al ombinatorial game known as the stablemarriage problem. We review lassi al results as the Gale-Shapley algorithmand present some re ent developments.Games ClubeMathMaria do Céu Soares, New University of LisbonMaria de Fátima Rodrigues, New University of LisbonNelson Chibeles Martins, New University of LisbonClubeMath is a Club that aims to show a di�erent side of Mathemati s,through fun and re reational a tivities, in order to stimulate skills and in-terest in this s ien e. Primarily dire ted to its members, whi h are studentsfrom basi and high s hools, ClubeMath also involves their parents and groupsof mathemati al tea hers in its a tivities. We will present some games thatwere developed for ClubeMath's a tivities. Namely we will talk about Bin-goMath, MathTrivial, Quem quer ser Mathmáti o?, Jogo da Glória Math-máti o, Sabes MathMais do que os teus Pais?, Assalto às MathMasmorrasand, �nally, the most re ent Tetr4Math, that we intend to develop for future ommer ialization. All of the aforementioned games were initially reatedfor use in ClubeMath's regular sessions but, meanwhile, they were alreadyapplied in di�erent basi and high s hools, either for re reational purposesor in the ontext of reviewing the mathemati al knowledge a quired in the lassroom.Winning Nim with Beatty and Fibona iMaria Joana Torres, University of MinhoIn this talk we des ribe how Fibona i representations play a role in deter-mining winning moves for Wytho�'s Nim very analogous to the role of binaryrepresentations in Bouton's Nim. We relate the winning moves for Wytho�'sNim with Beatty sequen es, so named after Beatty's beautiful theorem, �rstproposed as a problem in The Ameri an Mathemati al Monthly in 1926.Extreme Alphameti sMi hael KeithIn this talk we will dis uss the onstru tion and solution of new alphameti s(puzzles like SEND + MORE = MONEY, where the letters are to be re-pla ed with digits so that the arithmeti is valid) that stret h the boundariesof the form. Our results in lude base-10 addition problems with thousandsof distin t words, new re ords in doubly-true alphameti s (those of the form23

SEVEN + SEVEN + SIX = TWENTY), large alphameti s in prose andpoetry, and other "extreme" variations.The Mathemati s of Cartoons: A Brief SurveyNatália Bebiano, University of CoimbraJason Bolito, University of CoimbraF. Craveiro de Carvalho, University of CoimbraRoughly, a artoon is a short text and a drawing. The text is in isive, ironi or tragi , however, the text may not exist if the drawing is self-su� ient.The quality of the drawing is important, but not essential. Almost everynewspapers publish artoons, often politi al artoons, but, sometimes, theyarise in other ontexts, the s ienti� being a parti ular ase. In this paperwe intend to analyze some published artoons with Mathemati s as base,looking at some non-trivial examples.Games for Lazy but Intelligent ParentsRi hard Nowakowski, Dalhousie UniversityMany a long-su�ering parent has played the same game over and over withtheir o�spring. The kids want to win but not too easily else they get boredwhi h is even worse than playing the game for the parent. I'll present somegames and strategies that promise to keep the kids busy.Master lasses and Maths BuskingSara Santos, Royal Institution of Great BritainTwo extremes of maths ommuni ation: lasses for keen young mathemati- ians and maths as street entertainment for shoppers and strollers. Twomodels apparently so di�erent share a ommon point: exploit the innate hu-man uriosity. We des ribe the Royal Institution Mathemati s Master lassesfor Young People (established in 1981) and the new proje t Maths Buskingand explain how you an get involved to share your passion for mathemati s.Bodies Invisible in Several Dire tionsVera Rosh hina, CIMA-UE, University of ÉvoraAlexander Plakhov, University of AveiroWe onsider bodies moving in a rare�ed �ow of non-intera ting parti les (asin the theory of billiards or in geometri al opti s) and show that it is possi-ble to onstru t fra tal bodies invisible in 3 dire tions simultaneously. Thereare still many open questions, for example, what is the maximal number ofdire tions of invisibility, and whether it is possible to onstru t onne tedbodied invisible in 2 dire tions in 3 dimensional ase.24

5 Mailing ListAlda Carvalho ISEL a arvalho�dem.isel.ipl.ptAdelaide Carreira University of Lisbon arreira�f .ul.ptAlexandre Silva University of Lisbon ammmsilva�gmail. omAmália Cristina Corrente ES Rainha Santa Isabel de Estremoz amalia orrente�gmail. omAna Eliete University of Lisbon anaeliete�gmail. omAna Isabel Santos University of Évora aims�uevora.ptAna Júlia Martins University of Évora and ISEC anajuliamartins�sapo.ptAna Paula Silva ES Ca ilhas Tejo anaprfsilva�gmail. omAnabela Teixeira Museum of S ien e, University of Lisbon ateixeira�museus.ul.ptAndreia Hall University of Aveiro andreia.hall�ua.ptAntonieta Constantino ES de Mira�ores maria.a. . �gmail. omAntónio Manilhas EBI André de Resende de Évora antonio.manilhas�gmail. omAntónio Ma hiavelo University of Oporto ajma hiaf .up.ptCarlos Correia Ramos University of Évora r�uevora.ptCarlos Pereira dos Santos ISEC arlos.santos�ise .universitas.ptCarlota Dias High S hool Matias Aires arlotadia�gmail. omCarlota Simões University of Coimbra arlota�mat.u .ptCláudia Bi ho laudia. vb�gmail. omColin Wright Liverpool Mathemati al So iety juggler�solipsys. o.ukCristina Serpa University of Lisbon ristinaserpa�hotmail. omDores Ferreira University of Minho doresferreira�ie .uminho.ptEbrima Bojang Development Resear h Network info.derenet�gmail. omEdward Pegg Wolfram Resear h ed�mathpuzzle. omEvanina Martins ES Rainha Santa Isabel de Estremoz evanina.martins�gmail. omFilipe Papança Militar A ademy �lipe.papan a�gmail. omFilomena Carreira University of Lisbon �lomena arreira�sapo.ptHelder Pinto University of Lisbon hbmpinto1981�gmail. omHelena Sousa Melo University of Azores hmelo�ua .ptIlda Perez da Silva University of Lisbon isilva� ii.f .ul.ptInês Leitao University of Lisbon leitao.ines�gmail. omJoana Fernandes University of Lisbon joanatfernandes�hotmail. omJoão Cabral University of Azores j abral�ua .ptJoão Ma hado University of Évora jfnsma hado�gmail. omJoão Pedro Neto University of Lisbon jpn�di.f .ul.ptJoaquim Alves Gaspar CIHCT-FCUL, University of Lisbon alvesgaspar�net abo.ptJoaquim Euri o Nogueira New University of Lisbon jen�f t.unl.ptJohn Doe Development Resear h Network info.derenet�gmail. omJorge Bues u University of Lisbon jbues u�ptmat.f .ul.ptJorge Nuno Silva University of Lisbon jnsilva� al.berkeley.eduKeith Devlin Stanford University kdevlin�stanford.eduLennart Green World Champion lose-up/ ard Magi ian axland�telia. omLiliana Monteiro University of Lisbon lilianamonteiro�net.sapo.ptLuísa Carvalho University of Évora arvalh.luisa�gmail. omLuís Bernardino EB2,3 Dr. António da Costa Contreiras ebluisprof�gmail. omLuís Nunes University of Évora nunesdeal a er�gmail. omMaria do Céu Soares New University of Lisbon m s�f t.unl.ptMaria Joana Torres University of Minho jtorres�math.uminho.ptMarina Ca hu ho ES Rainha Santa Isabel de Estremoz marina a hu ho�sapo.ptMarta Duarte University of Lisbon marta.gab�gmail. omMi hael Keith mikek314�gmail. omNata ha Borrega University of Évora pixis934�hotmail. omNatália Bebiano University of Coimbra bebiano�mat.u .ptNoémia Simões ISEL and ISEG noemia erto�gmail. omPatrí ia Ferreira University of Lisbon patri iagferreira�hotmail. omPedro Pimentel de Sousa University of Lisbon pedropimenteldesousa�gmail. omRi ardo Matos University of Lisbon ri ardo.nuno.matos�gmail. omRi hard Nowakowski Dalhousie University rjn�mathstat.dal. aRita Magalhães University of Lisbon ritz.magalhaes�hotmail. omRuben Oliveira University of Évora ruben.oliveira17�hotmail. omRui Dias University of Évora ruidias666�gmail. omSandra Vinagre University of Évora smv�uevora.ptSandro Martins University of Lisbon s mm_91�hotmail. omSara Santos Royal Institution of Great Britain ssantos�ri.a .ukSusana Pereira University of Oporto s.simoes.pereira�gmail. omTatiana Ribeiro University of Lisbon tvbribeiro�gmail. omTeresa Iná io University of Lisbon ina ioteresa�hotmail. omTiago Santos Hirth University of Lisbon xztiago�hotmail. omValter Nunes University of Lisbon valterix�iol.ptVera Rosh hina CIMA-UE, University of Évora vera.rosh hina�gmail. omVeróni a Filipe Centro de Formação Pro�ssional de Tomar veroni a.�lipe� lix.pt25

27/04 - Wednesday 28/04 - Thursday 29/04 - Friday 30/04 - Saturday8:30 - 9:15 Registration9:15 - 9:30 Opening Session9:30 - 10:30 Re ently Solved Problems Master lasses and Maths Busking Games for Lazy but Intelligent Parents What if all middle s hool mathemati s edu ation were presentedEdward Pegg, Wolfram Resear h Sara Santos, Royal Institution of Great Britain Ri hard Nowakowski, Dalhousie University as re reational mathemati s? Keith Devlin, Standford University10:30 - 11:00 Bodies Invisible in Several Dire tions On a Certain Chaoti Game The shape of the sound: from bird singing to western musi One Noti es on the Creation and use of Pedagogi al Games in the Mathemati s LessonsVera Rosh hina, CIMA-UE, University of Évora Carlos Ramos, University of Évora Carlota Simões, University of Coimbra Ana Júlia Martins, University of Évora and ISEC11:00 - 11:30 Co�ee-Break Co�ee-Break Co�ee-Break Co�ee-Break11:30 - 12:00 Fooling around with Stable Marriages Using Empiri al Map Proje tions for Resear h, Edu ation and Fun The Mathemati s of Cartoons: A Brief Survey Winning Nim with Beatty and Fibona iJorge Bues u, University of Lisbon Joaquim Alves Gaspar, University of Lisbon Natália Bebiano, University of Coimbra Maria Joana Torres, University of Minho12:00 - 12:30 Syzygies played by elementary s hool students Some Math Problems with Trains and Railways Extreme Alphameti s Treason GameDores Ferreira, University of Minho Helder Pinto, University of Lisbon Mi hael Keith João Cabral, University of Azores12:30 - 13:00 Vogeler's Diagram and Rational Derived Polynomials Predominan e Game Sangaku Slimetrail for the Visual ImpairedAlda Carvalho, ISEL Helena Sousa Melo, University of Azores Antonieta Constantino, ES de Mira�ores Carlota Dias, High S hool Matias Aires13:00 - 14:30 Break for Lun h Break for Lun h Break for Lun h Break for Lun h14:30 - 15:30 Juggling: Theory and Pra ti e Magi Show The Last Geometri Argument in the History of PiColin Wright, Liverpool Mathemati al So iety Lennart Green, World Champion lose-up/ ard Magi ian Carlos Pereira dos Santos, ISEC15:30 - 16:00 Medieval Problems Mathemati al Quilts Games ClubeMathJoaquim Euri o Nogueira, New University of Lisbon Andreia Hall, University of Aveiro Maria do Céu Soares, New University of Lisbon16:00 - 16:30 Co�ee-Break Visit to Vila Viçosa Co�ee-Break Closing Session16:30 - 17:00 The Football Pool Problem Bo age and Mathemati sAntónio Ma hiavelo, University of Oporto Filipe Papança, Militar A ademy17:00 - 17:30 Shannon Swit hing Game and Variants Five Years of Ludus Asso iationIlda Perez, University of Lisbon Carlos Pereira dos Santos, ISEC17:30 - 18:00 A dynami al approa h to ne kla es and words Leonardo of Pisa (Fibona i)Cristina Serpa, University of Lisbon Keith Devlin, Standford University18:00 - 18:30 Mathemati al Cir us Break-TimeAdelaide Carreira, University of Lisbon18:30 - 19:00 Wine Tasting (Rota dos Vinhos do Alentejo)Break-Time19:00 - 19:30 Wel ome Co ktail (Évora Town Hall)19:30 - 20:00 Conferen e Dinner20:00 - 0:00