Professor Stewart’s Hoard of Mathematical Treasures ........................................... By the Same Author ConceptsofModernMathematics Game,Set,andMath DoesGodPlayDice? AnotherFineMathYou’veGotMeInto FearfulSymmetry(withMartinGolubitsky) Nature’sNumbers FromHeretoInfinity TheMagicalMaze Life’sOtherSecret Flatterland WhatShapeIsaSnowflake? TheAnnotatedFlatland(withEdwinA.Abbott) MathHysteria TheMayorofUglyville’sDilemma LetterstoaYoungMathematician HowtoCutaCake WhyBeautyIsTruth TamingtheInfinite ProfessorStewart’sCabinetofMathematicalCuriosities withJackCohen TheCollapseofChaos FigmentsofReality WhatDoesaMartianLookLike? Wheelers(sciencefiction) Heaven(sciencefiction) withTerryPratchettandJackCohen TheScienceofDiscworld TheScienceofDiscworldII:TheGlobe TheScienceofDiscworldIII:Darwin’sWatch Professor Stewart’s Hoard of Mathematical Treasures ........................................... Ian Stewart A Member of the Perseus Books Group New York Hardcover first published in Great Britain in 2009 by PROFILE BOOKS LTD 3A Exmouth House Pine Street London WEC1R 0JH www.profilebooks.com Paperback first published in 2010 by Basic Books, A Member of the Perseus Books Group Copyright © Joat Enterprises 2009 Text design by Sue Lamble Typeset in Stone Serif by Data Standards Ltd, Frome, Somerset. Printed and bound in Britain by Clays, Bungay, Suffolk. The moral authority of the author has been asserted. All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodiedin critical articles and reviews. For information, address Basic Books, 387 Park Avenue South, New York, NY 10016-8810. Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the PerseusBooks Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail [email protected]. Hardcover ISBN: 978-1-84668-292-6 Paperback ISBN: 978-0-465-01775-1 10 9 8 7 6 5 4 3 2 1 Contents ........................................... Acknowledgements//xi SecondDrawerDown ... //1 CalculatorCuriosity1//7 YearTurnedUpsideDown//7 LucklessLovelornLilavati//8 SixteenMatches//9 SwallowingElephants//9 MagicCircle//10 Dodgem//10 Press-the-Digit-ation//11 SecretsoftheAbacus//13 Redbeard’sTreasure//16 Hexaflexagons//18 WhoInventedtheEqualsSign?//20 StarsandSnips//20 BytheNumbersofBabylon//21 MagicHexagons//25 TheCollatz–Syracuse–UlamProblem//26 TheJeweller’sDilemma//29 WhatSeamusDidn’tKnow//30 WhyToastAlwaysFallsButtered-SideDown//31 vi//Contents TheButteredCatParadox//31 Lincoln’sDog//32 Whodunni’sDice//33 AFlexiblePolyhedron//33 ButWhatAboutConcertinas?//34 TheBellowsConjecture//35 DigitalCubes//38 NothingWhichAppealsMuchtoaMathematician//38 WhatIstheAreaofanOstrichEgg?//40 OrderintoChaos//43 BigNumbers//44 TheDrowningMathematician//49 MathematicalPirates//49 TheHairyBallTheorem//52 CupsandDowns//53 SecretCodes//56 When2+2=0//59 SecretCodesThatCanBeMadePublic//62 CalendarMagic//66 MathematicalCats//67 TheRuleofEleven//70 DigitalMultiplication//70 CommonKnowledge//71 PickledOnionPuzzle//73 GuesstheCard//74 AndNowwithaCompletePack//75 Halloween=Christmas//75 EgyptianFractions//76 TheGreedyAlgorithm//80 HowtoMoveaTable//83 RectanglingtheSquare//83 Newton,byByron//84 XMarkstheSpot//84 Contents//vii Whatever’stheAntimatter?//86 HowtoSeeInsideThings//91 MathematiciansMusingAboutMathematics//96 Wittgenstein’sSheep//98 LeaningTowerofPizza//99 PieThagoras’sWorld-FamousMinceπs//103 DiamondFrame//104 PourRelations//105 Alexander’sHornedSphere//105 TheSacredPrincipleofMat//107 PerfectlyAbundantlyAmicablyDeficient//109 TargetPractice//111 JustaPhaseI’mGoingThrough//112 ProofTechniques//114 SecondThoughts//116 HowDudeneyCookedLoyd//116 CookingwithWater//117 CelestialResonance//118 CalculatorCuriosity2//123 WhichisBigger?//123 SumsThatGoOnForEver//123 TheMostOutrageousProof//126 ColoradoSmithandtheSolarTemple//129 WhyCan’tIAddFractionsLikeIMultiplyThem?//131 Farey,Farey,QuiteContrary//131 PoolingResources//133 WelcometotheRep-TileHouse//133 CookingonaTorus//135 TheCatalanConjecture//136 TheOriginoftheSquareRootSymbol//138 PleaseBearwithMe//139 TheHamSandwichTheorem//139 CricketonGrumpius//143 viii//Contents TheManWhoLovedOnlyNumbers//144 TheMissingPiece//145 TheOtherCoconut//145 WhatDoesZeno?//146 PiecesofFive//153 PiintheSky//154 TheCuriousIncidentoftheDog//155 MathematicsMadeDifficult//155 AWeirdFactaboutEgyptianFractions//157 AFourColourTheorem//157 SerpentofPerpetualDarkness//158 WhatAretheOdds?//159 APottedHistoryofMathematics//160 TheShortestMathematicalJokeEver//163 GlobalWarmingSwindle//164 NametheCards//171 WhatIsPointNineRecurring?//172 GhostofaDepartedQuantity//174 NiceLittleEarner//176 APuzzleforLeonardo//176 CongruentNumbers//177 Present-MindedSomewhereElse//179 It’sAboutTime//180 DoIAvoidKangaroos?//180 TheKleinBottle//181 AccountingtheDigits//185 MultiplyingwithSticks//186 AsLongasIGazeonLaplacianSunrise//190 AnotherTakeonMathematicalCats//191 BorderedPrimeMagicSquare//192 TheGreen–TaoTheorem//192 Peaucellier’sLinkage//195 ABetterApproximationtoπ//196 Contents//ix StrictlyforCalculusBuffs//197 TheStatueofPallasAthene//198 CalculatorCuriosity3//199 CompletingtheSquare//200 TheLookandSaySequence//200 Non-MathematiciansMusingAboutMathematics//201 Euler’sConjecture//203 TheMillionthDigit//203 PiraticalPathways//204 TrainsThatPassintheSiding//205 PleaseMakeYourselfClear//205 Squares,ListsandDigitalSums//206 Hilbert’sHit-List//206 MatchTrick//211 WhichHospitalShouldClose?//212 HowtoTurnaSphereInsideOut//213 APieceofStringWalkedintoaBar...//218 SlicingtheCake//218 TheOriginoftheSymbolforPi//219 HallofMirrors//219 GreekandTrojanAsteroids//220 SlidingCoins//223 BeatThat!//223 Euclid’sPuzzle//224 TheInfiniteMonkeyTheorem//225 MonkeysAgainstEvolution//226 UniversalLetterofReference//228 SnakesandAdders//228 PowerfulCrossnumber//230 MagicHandkerchiefs//231 ABluffer’sGuidetoSymmetry//232 DigitalCenturyRevisited//236 AnInfinityofPrimes//237