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Winning Ways: for Your Mathematical Plays Winning Ways for Your Mathematical Plays, Volume 2 Winning Ways for Your Mathematical Plays Ɔ Volume 2, Second Edition Elwyn R. Berlekamp, John H. Conway, Richard K. Guy A K Peters Natick, Massachusetts Editorial, Sales, and Customer Service Office A K Peters, Ltd. 63 South Avenue Natick, MA 01760 Copyright © 2003 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any informa- tion storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Berlekamp, Elwyn R. Winning Ways for your mathematical plays / Elwyn Berlekamp, John H. Conway, Richard Guy.--2nd ed. p. cm. Includes bibliographical references and index. ISBN 1-56881-130-6 (v. 1) – ISBN 1-56881-142-X (v. 2) – ISBN 1-56881-143-8 (v. 3) – ISBN 1-56881-144-6 (v. 4) (alk.paper) 1. Mathematical recreations. I. Conway, John Horton. II. Guy Richard K. III. Title. QA95 .B446 2000 739.7’4--dc21 00-048541 Printed in Canada 07 06 05 04 03 10 9 8 7 6 5 4 3 2 1 To Martin Gardner who has brought more mathematics to more millions than anyone else Elwyn Berlekamp was born in Dover, Ohio, on September 6, 1940. He has been Professor of Mathematics and of Electrical Engineering/Com- puter Science at UC Berkeley since 1971. He has also been active in several technology business ventures. In addition to writing many jour- nal articles and several books, Berlekamp also has 12 patented inven- tions, mostly dealing with algorithms for synchronization and error correction. He is a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences. From 1994 to 1998, he was chairman of the board of trust- ees of the Mathematical Sciences Research Institute (MSRI). John H. Conway was born in Liverpool, England, on December 26, 1937. He is one of the preeminent theorists in the study of finite groups and the mathematical study of knots, and has written over 10 books and more than 140 journal articles. Before joining Princeton University in 1986 as the John von Neumann Distinguished Professor of Mathematics, Conway served as profes- sor of mathematics at Cambridge University, and remains an hon- orary fellow of Caius College. The recipient of many prizes in re- search and exposition, Conway is also widely known as the inventor of the Game of Life, a computer simulation of simple cellular “life,” governed by remarkably simple rules. Richard Guy was born in Nuneaton, England, on September 30, 1916. He has taught mathematics at many levels and in many places— England, Singapore, India, and Canada. Since 1965 he has been Pro- fessor of Mathematics at the University of Calgary, and is now Fac- ulty Professor and Emeritus Professor. The university awarded him an Honorary Degree in 1991. He was Noyce Professor at Grinnell College in 2000. He continues to climb mountains with his wife, Louise, and they have been patrons of the Association of Canadian Mountain Guides’ Ball and recipients of the A. O. Wheeler award for Service to the Alpine Club of Canada. Contents Preface to Second Edition xiii Preface xiv Change of Heart xvii 9 IfYouCan’tBeat’Em,Join’Em! 277 AlltheKing’sHorses..................................... 277 WeCanJoinAnyGames.................................. 278 HowRemoteIsaHorse?................................... 278 WhatIftheFirstHorsetoGetStuckWins?........................ 281 ASlightlySlowerJoin.................................... 283 MovingHorsesImpartially.................................. 283 CuttingEveryCake..................................... 284 Eatcakes............................................ 286 WhentoPutYourMoneyontheLastHorse....................... 286 SlowHorsesJointheAlso-Rans............................... 286 LetThemEatCake..................................... 289 All the King’s Horses on a Quarter-InfiniteBoard..................... 292 CuttingYourCakesandEatingThem........................... 292 ReferencesandFurtherReading............................... 298 10 Hot Battles Followed by Cold Wars 299 ColdGames—NumbersAreStillNumbers......................... 300 HotGames—TheBattleIsJoined!............................. 300 Tolls,TimersandTallies................................... 301 WhichIstheBestOption?................................. 303 HotPositions......................................... 304 ColdPositions........................................ 304 TepidPositions........................................ 306 ATepidGame........................................ 308 SelectBoysandGirls..................................... 310 Mrs.Grundy......................................... 310 How to Play Mis`ereUnionsofPartizanGames...................... 312 UrgentUnions(ShotgunWeddings?)............................ 312 Predeciders—OverridersandSuiciders........................... 312 ix xContents ♥ Falada............................................. 312 TwoMoreFaladaGames.................................. 320 BakedAlaska......................................... 321 AFelicitousFaladaField.................................. 324 The Rules for Tallies on InfiniteTolls........................... 325 TimeMayBeShortherthanYouThink!.......................... 326 11 Games Infinite and Indefinite 327 InfiniteEnders........................................ 329 The InfiniteOrdinalNumbers................................ 329 OtherNumbers........................................ 330 InfiniteNim.......................................... 330 The InfiniteSprague-GrundyandSmithTheories..................... 333 SomeSuperheavyAtoms................................... 334 Fixed,MixedandFree.................................... 335 Onsides and Offsides,UpsumsandDownsums....................... 336 Stoppers............................................ 337 on, off and dud ....................................... 337 How Big Is on?........................................ 338 It’sBiggerthanAllofThem!................................ 338 SidlingTowardsaGame................................... 338 SidlingPicksSides...................................... 340 StoppersHaveOnlyOneSide................................ 340 ’Tis!-’Tisn!-’Tis!-’Tisn!-..................................... 342 LoopyHackenbush...................................... 343 DisentanglingLoopyHackenbush.............................. 343 Loopily InfiniteHackenbush................................. 344 Sisyphus............................................ 346 LivingwithLoops...................................... 348 ComparingLoopyGames.................................. 348 TheSwivelChairStrategy.................................. 349 StoppersAreNice...................................... 350 PlumtreesAreNicer!..................................... 353 TakingCareofPlumtrees.................................. 354 WorkingWithUpsumsandDownsums........................... 355 on, off and hot ........................................ 356 ASummaryofSomeSumProperties............................ 357 TheHouseofCards..................................... 357 TheDegreeofLoopiness................................... 361 ClassesandVarieties..................................... 362 NoHighway.......................................... 364 BackslidingToads-and-Frogs................................ 367 Bach’sCarousel........................................ 369 Getting on inCheckers................................... 370 ProofoftheSidlingTheorem................................ 371 AnswertoExerciseOne................................... 374 tis and tisn .......................................... 374 upon ............................................. 375 Contents xi ♥ BackslidingToads-and-Frogs................................ 375 KOsinGo........................................... 377 ReferencesandFurtherReading............................... 377 12 Games Eternal – Games Entailed 379 FairSharesandVariedPairs................................ 379 HowSoonCanYouWin?.................................. 381 There May Be Open Positions ( -Positions)........................ 382 Adders-and-Ladders.....................................O 386 JustHowLoopyCanYouGet?............................... 391 CorrallAutomotiveBettermentScheme.......................... 391 SharingOutOtherKindsofNut.............................. 393 FairSharesandUnequalPartners.............................. 394 SweetsandNuts,andMaybeaDate?........................... 395 TheAdditionalSubtractionGames............................. 395 Horsefly ............................................ 395 SelectiveandSubselectiveCompoundsofImpartialGames................ 396 EntailingMoves........................................ 396 SunnyandLoonyPositions................................. 397 CalculatingwithEntailedValues.............................. 398 NimwithEntailingMoves.................................. 400 Goldbach’sNim........................................ 401 WytQueenswithTrains................................... 402 AddingTailstoPrimandDim............................... 404 ComplimentingMoves.................................... 405 On-the-Rails......................................... 407 De Bono’s L-Games..................................... 408 ProvingtheOutcomeRulesforLoopyPositions...................... 408 FairSharesandUnequalPartners.............................. 410 WereYourWaysWinningEnough?............................. 410 DidYouMoveFirstinHorsefly?.............................
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