Bicycle or Unicycle？_A Collection of Intriguing Mathematical Puzzles

305 Pages·2020·14.709 MB·English

by  Stan Wagon, Daniel J. Velleman

AMS/MAA PROBLEM BOOKS VOL36
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DANIEL J . VELLEMAN STAN WAGON
A Collection of Intriguing
Mathematical Puzzles
•
Anlmprelnt :• . •
PRE ___. ofthe AMERICAN
\ ~: MATHEMATICAL
•.~ S0CIETY
ams.org.
Bicycle or Unicycle?
A Collection of Intriguing
Mathematical Puzzles
ams.org.
ams.org.
I
AMS/MAA PROBLEM BOOKS
VOL
36
Bicycle or Unicycle?
A Collection of Intriguing
Mathematical Puzzles
Daniel J. Velleman
Stan Wagon
•••
Anlmprlnt • ••
e::•
PRESS ofthe AMERICAN
4t!: MATI-IEMATICAL
•:•. SOCIETY
•
•••
MAAProblemBooksEditorialBoard
GailS.Nelson,Editor
AdamH.Berliner JenniferRocheBowen
MichelleL.Ghrist GregOman
EricR.Westlund
2010MathematicsSubjectClassification.Primary00A07,00A08.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/prb-36
LibraryofCongressCataloging-in-PublicationData
Names:Velleman,DanielJ.,author.|Wagon,S.,author.
Title:Bicycleorunicycle?:Acollectionofintriguingmathematicalpuzzles/DanielJ.Velleman,Stan
Wagon.
Description:Providence,RhodeIsland:MAAPress,animprintoftheAmericanMathematicalSoci-
ety,[2020]|Series:Problembooks;volume36|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2020003232|ISBN9781470447595(paperback)|ISBN9781470457020(ebook)
Subjects:LCSH:Mathematicalrecreations.|Mathematics–Problems,exercises,etc.|AMS:General
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neousspecifictopics–Recreationalmathematics.
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Contents
Preface xiii
Part1 Problems 1
1 CanaBicycleSimulateaUnicycle? 3
1 BicycleorUnicycle? 3
2 Geometry 5
2 TrisectingtoBenefitSociety 5
3 TenBottlesofWine 5
4 IntotheWoods 5
5 TheAttractionoftheGoldenRatio 6
6 ReflectonThis 6
7 SkewedPizza 7
8 Four-RegularSquares 7
9 VennSymmetry 8
10 ACrosscutQuadrilateral 9
11 TheLegacyofH.G.Wells 9
12 TilingSurprise 9
13 AWell-BalancedClock 11
14 OntheLevel 11
15 APolyhedralPuzzle 11
16 WiggleRoom 12
17 DrinkMe 12
18 Right-AngledPolygons 12
19 ARollingParabola 12
20 RocketScience 13
21 TheIcingontheCake 14
22 MoreCake 14
3 NumberTheory 15
23 PowerMatching 15
v
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vi Contents
24 Triplets 15
25 TheMysteriousSeventeenthDivisor 15
26 TrueorFalse? 15
27 AnExponentialDiophantineProblem 16
28 APrimeCharacterization 16
29 TwoSumsandManyDifferences 16
30 ReciprocalstoSquares 16
31 Nondivisibilityby11 16
32 EquallyPowerfulSplits 16
33 TriplinginTwoSteps 17
34 RootClosure 17
35 ADoublePowerLeadstoaSumofTwoSquares 17
36 PrimeSubsets 17
37 TheIncredibleShrinkingSuperpowers 17
38 SpecialNumbers 19
4 Combinatorics 21
39 PowerupYourRadio 21
40 AParkingPuzzle 21
41 ABlackandWhiteIssue 22
42 AChoiceProblem 22
43 WaterWave 22
44 AnAcceptableCommittee 22
45 ACompatibleCommittee 23
46 UsingBlocksandChainstoUnlockaSafe 23
47 AUniversalSetofDirections 24
48 ACompetitionProblemAboutaCompetitionProblem 24
5 Probability 25
49 TheImportanceofIrrelevantInformation 25
50 CreepingAnts 25
51 RolltheDice 25
52 ConditionedThrowsofaDie 26
53 WhereAretheRoundedPowersofTwo? 26
54 TheHolyGameofPoker 26
55 AShrinkingRandomWalk 27
56 BINGO! 27
57 HowMuchisaPennyWorth? 27
58 AHistoricallyInterestingTruth-TellerProblem 28
59 MontyHallRevisited 28
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Contents vii
6 Calculus 31
60 IntegratingaBaseChange 31
61 AT-ShirtGun 31
62 ShootingRange 31
63 ConvergentRationalEnumeration 32
64 AFunctionalEquation 32
65 TwoEerieRecursions 32
66 StabilizingaCylinder 32
67 𝜋Coincidence? 32
68 TheChaseIsOn 32
7 AlgorithmsandStrategy 33
69 Where’sBob? 33
70 It’saHorseRace 33
71 YourTwoBestShots 33
72 DeterminetheMartianMajority 34
73 MajorityRules 34
74 GoingforGold 34
75 TheMensaCorrectionalInstitute 35
76 MatchingHats 35
77 OneHatTooMany 35
78 ThePrisonersMustAgree 36
79 HowtoUseIrrelevantInformation 37
80 FlippingPennies 37
81 BattleshipDestruction 37
82 RealBattleshipDestruction 38
83 AVeryLocalMaximum 38
84 DetectingaBlackHole 38
85 Pablito’sSolitaire 38
86 AretheCoinsAuthentic? 39
87 WebSiteAnalysis 39
88 FindtheCar,andtheCarKey 40
89 MagicCoins 40
90 RussianCards 40
91 TheGenerousAutomatedTellerMachine 41
8 Miscellaneous 43
92 ASelf-DescriptiveCrossword 43
93 SeriousImplications 43
94 HermioneGrangerandtheDeadlyBottles 43
95 ForbiddenPolynomials 45
96 APolynomialScandal 45
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viii Contents
97 Pascal’sDeterminant 46
98 PainsofImperfectGlass 46
99 OntheHighway 46
100 ARunningMystery 47
101 TheStarofThisProblemisAddition 47
102 Don’tGetYourWiresCrossed 47
103 WeightLossThroughJuggling 47
104 CantorSetArithmetic 48
105 TheMiracleoftheCollidingBlocks 49
Part2 Solutions 51
1 CanaBicycleSimulateaUnicycle? 53
1 BicycleorUnicycle? 53
2 Geometry 67
2 TrisectingtoBenefitSociety 67
3 TenBottlesofWine 67
4 IntotheWoods 68
5 TheAttractionoftheGoldenRatio 69
6 ReflectonThis 70
7 SkewedPizza 72
8 Four-RegularSquares 74
9 VennSymmetry 79
10 ACrosscutQuadrilateral 80
11 TheLegacyofH.G.Wells 80
12 TilingSurprise 81
13 AWell-BalancedClock 84
14 OntheLevel 86
15 APolyhedralPuzzle 89
16 WiggleRoom 89
17 DrinkMe 95
18 Right-AngledPolygons 98
19 ARollingParabola 101
20 RocketScience 102
21 TheIcingontheCake 104
22 MoreCake 106
3 NumberTheory 109
23 PowerMatching 109
24 Triplets 111
25 TheMysteriousSeventeenthDivisor 111
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