The Mathematics of Various Entertaining Subjects Volume 2: Research in Games, Graphs, Counting, and Complexity,

409 Pages·2017·2.28 MB·English

by  Jennifer Elaine

TheMathematicsofVariousEntertainingSubjects
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T M V
HE ATHEMATICS OF ARIOUS
E S
NTERTAINING UBJECTS
Volume 2
RESEARCH IN GAMES, GRAPHS,
COUNTING, AND COMPLEXITY
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EDITEDBY
Jennifer Beineke & Jason Rosenhouse
WITHAFOREWORDBYRONGRAHAM
NationalMuseumofMathematics,NewYork • PrincetonUniversityPress,PrincetonandOxford
Copyright(cid:2)c 2017byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,
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press.princeton.edu
InassociationwiththeNationalMuseumofMathematics,
11East26thStreet,NewYork,NewYork10010
Jacketart:Toprow(lefttoright)Fig.1:CourtesyofEricDemaineand
WilliamS.Moses.Fig.2:CourtesyofAvivAdler,ErikDemaine,AdamHesterberg,
QuanquanLiu,andMikhailRudoy.Fig.3:CourtesyofPeterWinkler.
Middlerow(lefttoright)Fig.1:CourtesyofErikD.Demaine,MartinL.Demaine,
AdamHesterberg,QuanquanLiu,RonTaylor,andRyuheiUehara.Fig.2:Courtesyof
RobertBosch,RobertFathauer,andHenrySegerman.Fig.3:Courtesyof
JasonRosenhouse.Bottomrow(lefttoright)Fig.1:CourtesyofNoamElkies.
Fig.2:CourtesyofRichardK.Guy.Fig.3:CourtesyofJillBigleyDunham
andGwynWhieldon.
Excerptfrom“Macavity:TheMysterCat”fromOldPossum’sBookofCats
byT.S.Eliot.Copyright1939byT.S.Eliot.Copyright(cid:2)c Renewed1967by
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LibraryofCongressCataloging-in-PublicationData
Names:Beineke,JenniferElaine,1969–editor.|Rosenhouse,Jason,editor.
Title:Themathematicsofvariousentertainingsubjects:researchingames,graphs,
counting,andcomplexity/editedbyJenniferBeineke&JasonRosenhouse;with
aforewordbyRonGraham.Description:Princeton:PrincetonUniversityPress;
NewYork:PublishedinassociationwiththeNationalMuseumofMathematics,
[2017]|Copyright2017byPrincetonUniversityPress.|Includesbibliographical
referencesandindex.
Identifiers:LCCN2017003240|ISBN9780691171920(hardcover:alk.paper)
Subjects:LCSH:Mathematicalrecreations-Research.
Classification:LCCQA95.M368742017|DDC793.74–dc23LCrecord
availableathttps://lccn.loc.gov/2017003240
BritishLibraryCataloging-in-PublicationDataisavailable
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13579108642
Contents
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ForewordbyRonGraham vii
PrefaceandAcknowledgments xi
PARTIPUZZLESANDBRAINTEASERS
1
TheCyclicPrisoners 3
PeterWinkler
2
DragonsandKasha 11
TanyaKhovanova
3
TheHistoryandFutureofLogicPuzzles 23
JasonRosenhouse
4
TheTowerofHanoiforHumans 52
PaulK.Stockmeyer
5
Frenicle’s880MagicSquares 71
JohnConway,SimonNorton,andAlexRyba
PARTIIGEOMETRYANDTOPOLOGY
6
ATriangleHasEightVerticesButOnlyOneCenter 85
RichardK.Guy
7
EnumerationofSolutionstoGardner’sPaperCutting
andFoldingProblem 108
JillBigleyDunhamandGwynethR.Whieldon
8
TheColorCubesPuzzlewithTwoandThreeColors 125
EthanBerkove,DavidCervantes-Nava,DanielCondon,
AndrewEickemeyer,RachelKatz,andMichaelJ.Schulman
9
TangledTangles 141
ErikD.Demaine,MartinL.Demaine,AdamHesterberg,
QuanauanLiu,RonTaylor,andRyuheiUehara
vi • Contents
PARTIIIGRAPHTHEORY
10
MakingWalksCount:FromSilentCirclesto
HamiltonianCycles 157
MaxA.AlekseyevandGérardP.Michon
11
Duels,Truels,Gruels,andSurvivaloftheUnfittest 169
DominicLanphier
12
Trees,Trees,SoManyTrees 195
AllenJ.Schwenk
13
CrossingNumbersofCompleteGraphs 218
NoamD.Elkies
PARTIVGAMESOFCHANCE
14
NumericallyBalancedDice 253
RobertBosch,RobertFathauer,andHenrySegerman
15
ATROUBLE-someSimulation 269
GeoffreyD.Dietz
16
ASequenceGameonaRouletteWheel 286
RobertW.Vallin
PARTVCOMPUTATIONALCOMPLEXITY
17
MultinationalWarIsHard 301
JonathanWeed
18
ClickomaniaIsHard,EvenwithTwoColorsandColumns 325
AvivAdler,ErikD.Demaine,AdamHesterberg,QuanquanLiu,
andMikhailRudoy
19
ComputationalComplexityofArrangingMusic 364
ErikD.DemaineandWilliamS.Moses
AbouttheEditors 379
AbouttheContributors 381
Index 387
Foreword
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RonGraham
recreation—somethingpeopledoto
relaxorhavefun.
—Merriam–WebsterDictionary
One of the strongest human instincts is the overwhelming urge to “solve
puzzles.”Whetherthismeanshowtomakefire,avoidbeingeatenbywolves,
keepdryintherain,orpredictsolareclipses,these“puzzles”havebeenwith
ussincebeforecivilization.Ofcourse,peoplewhowerebetteratsuccessfully
dealing with such problems had a better chance of surviving, and then,
as a consequence, so did their descendants. (A current (fictional) solver of
problems like this is the character played by Matt Damon in the recent film
TheMartian).
On a more theoretical level, mathematical puzzles have been around for
thousandsofyears.ThePalimpsest ofArchimedescontainsseveralpagesde-
votedtotheso-calledStomachion,ageometricalpuzzleconsistingoffourteen
polygonalpieceswhicharetobearrangedintoa12×12square.Itisbelieved
that the problem given was to enumerate the number of different ways this
couldbedone,butsinceanumberofthepagesofthePalimpsestaremissing,
wearenotquitesure.
Itiswidelyacknowledgedbynowthatmanyrecreationalpuzzleshaveled
toquitedeepmathematicaldevelopmentsasresearchersdelvedmoredeeply
intosomeoftheseproblems.Forexample,theexistenceofPythagoreantriples,
suchas
32+42 =52,
andquarticquadruples,suchas
26824404+153656394+187967604 =206156734,
ledtoquestions,suchaswhether
xn+yn =zn
couldeverholdforpositiveintegersx,y,andzwhenn≥3.(Theanswer:No!
ThiswasAndrewWiles’resolutionofFermat’sLastTheorem,whichspurred
thedevelopmentofevenmorepowerfultoolsforattackingevenmoredifficult
viii • Foreword
TABLE1.
Numbersexpressibleintheformn=6xy±x±y
x y 6xy+x+y 6xy+x−y 6xy−x+y 6xy−x−y
1 1 8 6 6 4
2 1 15 13 11 9
2 2 28 24 24 20
3 1 22 20 16 14
3 2 41 37 35 31
3 3 60 54 54 48
4 1 29 27 21 19
. . . . . .
. . . . . .
. . . . . .
questions.)Similarstoriescouldbetoldinavarietyofotherareas,suchasthe
analysisofgamesofchanceintheMiddleAgesleadingtothedevelopmentof
probabilitytheory,andthestudyofknotsleadingtofundamentalworkonvon
Neumannalgebras.
In 1900, at the International Congress of Mathematicians in Paris, the
legendarymathematicianDavidHilbertgavehiscelebratedlistoftwenty-three
problemswhichhefeltwouldkeepthemathematiciansbusyfortheremainder
of the century. He was right! Many of these problems are still unsolved.
(Actually, he only mentioned eight of the problems during his talk. The full
listoftwenty-threewasonlypublishedlater.)Inthatconnection,Hilbertalso
wroteabouttheroleofproblemsinmathematics.Paraphrasing,he saidthat
problemsarethecoreofanymathematicaldiscipline.Itiswithproblemsthat
youcan“testthetemperofyoursteel.”However,itisoftendifficulttojudge
thedifficulty(orimportance)ofaparticularprobleminadvance.Letmegive
twoofmyfavoriteexamples.
Problem1. Considerthesetofpositiveintegersnwhichcanberepresentedas
n=6xy±x±y,
wherex ≥ y ≥0.SomesuchnumbersaredisplayedinTable1.
Itseemslikemostofthesmallnumbersoccurinthetable,althoughsomeare
missing.Thelistofthemissingnumbersbegins
{1,2,3,5,7,10,12,17,18,23,...}.
Arethereinfinitelymanynumbersmthatarenotinthetable?
Iwillgivetheanswerattheend.Hereisanotherproblem.
Foreword • ix
Problem 2. A well-studied function in number theory is the divisor function
d(n),whichdenotesthesumofthedivisorsoftheintegern.Forexample,
d(12)=1+2+3+4+6+12=28,
and
d(100)=1+2+4+5+10+20+25+50+100=217.
Anothercommonfunctioninmathematicsistheharmonicnumber H(n).Itis
definedby
(cid:2)n
1
H(n)= .
k
k=1
In other words, H(n) is the sum of the reciprocals of the first n integers. Is it
truethat
d(n)≤ H(n)+eH(n)logH(n),
forn≥1?
Howhardcouldthisbe?Actually,prettyhard(orsoitseems!).
Readers of this volume will find an amazing assortment of brainteasers,
challenges,problems,and“puzzles”arisinginavarietyofmathematical(and
non-mathematical)domains.Andwhoknowswhethersomeoftheseproblems
willbetheacornsfromwhichmightymathematicaloakswillsomedayemerge!
Asfortheproblems,theanswertoeachisthatnooneknows!
ForProblem1,eachnumbermthatismissingfromthetablecorrespondsto
apairoftwinprimes6m−1,6m+1.Furthermore,everypairoftwinprimes
(except 3 and 5) occur this way. Recall, a pair of twin primes is a set of two
primenumberswhichdifferbytwo.Thus,Problem1isreallyaskingwhether
there are infinitely many pairs of twin primes. As Paul Erdo˝s liked to say,
“Everyright-thinkingpersonknowstheanswerisyes,”butsofarnoonehas
been able to prove this. It is known that there exist infinitely many pairs of
primeswhichdifferbyatmost246,theestablishmentofwhichwasactuallya
majorachievementinitself!
ForProblem2,itisknownthattheanswerisyesifandonlyiftheRiemann
Hypothesis holds! As I said, this appears to be a rather difficult problem
at present (to say the least). It appears on the list of the Clay Millennium
Problems,witharewardonofferofonemilliondollars.Goodluck!