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http://dx.doi.org/10.1090/psapm/043 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied Mathematics Volume 43 COMBINATORIAL GAMES Edited by Richard K Guy [Columbus, Ohio, August 1990) Volume 42 CRYPTOLOGY AND COMPUTATIONAL NUMBER THEORY Edited by C. Pomerance {Boulder, Colorado, August 1989) Volume 41 ROBOTICS Edited by R. W. Brockett (Louisville, Kentucky, January 1990) Volume 40 MATRIX THEORY AND APPLICATIONS Edited by Charles R. Johnson (Phoenix, Arizona, January 1989) Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L. Devaney and Linda Keen (Providence, Rhode Island, August 1988) Volume 38 COMPUTATIONAL COMPLEXITY THEORY Edited by Juris Hartmanis (Atlanta, Georgia, January 1988) Volume 37 MOMENTS IN MATHEMATICS Edited by Henry J. Landau (San Antonio, Texas, January 1987) Volume 36 APPROXIMATION THEORY Edited by Carl de Boor (New Orleans, Louisiana, January 1986) Volume 35 ACTUARIAL MATHEMATICS Edited by Harry H. Panjer (Laramie, Wyoming, August 1985) Volume 34 MATHEMATICS OF INFORMATION PROCESSING Edited by Michael Anshel and William Gewirtz (Louisville, Kentucky, January 1984) Volume 33 FAIR ALLOCATION Edited by H. Peyton Young (Anaheim, California, January 1985) Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS Edited by R. W. McKelvey (Eugene, Oregon, August 1984) Volume 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 1983) Volume 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983) Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton (San Francisco, California, January 1981) Volume 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982) Volume 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982) Volume 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981) Volume 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S. I. Gass (Duluth, Minnesota, August 1979) Volume 24 GAME THEORY AND ITS APPLICATIONS Edited by W. F. Lucas (Biloxi, Mississippi, January 1979) PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS Volume 23 MODERN STATISTICS: METHODS AND APPLICATIONS Edited by R. V. Hogg {San Antonio, Texas, January 1980) Volume 22 NUMERICAL ANALYSIS Edited by G. H. Golub and J. Oliger {Atlanta, Georgia, January 1978) Volume 21 MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF ENERGY Edited by P. D. Lax (San Antonio, Texas, January 1976) Volume 20 THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH AND EDUCATION Edited by J. P LaSalle {University of Montana, August 1973) Volume 19 MATHEMATICAL ASPECTS OF COMPUTER SCIENCE Edited by J. T Schwartz {New York City, April 1966) Volume 18 MAGNETO-FLUID AND PLASMA DYNAMICS Edited by H. Grad {New York City, April 1965) Volume 17 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS Edited by R. Finn {New York City, April 1964) Volume 16 STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND ENGINEERING Edited by R. Bellman {New York City, April 1963) Volume 15 EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND MATHEMATICS Edited by N. C Metropolis, A. H. Taub, J. Todd, and C B. Tompkins {Atlantic City and Chicago, April 1962) Volume 14 MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES Edited by R. Bellman {New York City, April 1961) Volume 13 HYDRODYNAMIC INSTABILITY Edited by R. Bellman, G. Birkhoff and C C Lin {New York City, April I960) Volume 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS Edited by R. Jakobson {New York City, April I960) Volume 11 NUCLEAR REACTOR THEORY Edited by G. Birkhoff and E. P. Wigner {New York City, April 1959) Volume 10 COMBINATORIAL ANALYSIS Edited by R. Bellman and M. Hall, Jr. {New York University, April 1957) Volume 9 ORBIT THEORY Edited by G. Birkhoff and R. E. Langer {Columbia University, April 1958) Volume 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS Edited by L. M. Graves {University of Chicago, April 1956) Volume 7 APPLIED PROBABILITY Edited by L. A. MacColl {Polytechnic Institute of Brooklyn, April 1955) Volume 6 NUMERICAL ANALYSIS Edited by J. H. Curtiss {Santa Monica City College, August 1953) Volume 5 WAVE MOTION AND VIBRATION THEORY Edited by A. E. Heins {Carnegie Institute of Technology, June 1952) Volume 4 FLUID DYNAMICS Edited by M. H. Martin {University of Maryland, June 1951) PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS Volume 3 ELASTICITY Edited by R. V. Churchill (University of Michigan, June 1949) Volume 2 ELECTROMAGNETIC THEORY Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948) Volume 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947) AMS SHORT COURSE LECTURE NOTES Introductor y Surve y Lecture s publishe d a s a subserie s of Proceeding s of Symposi a in Applie d Mathematic s PROCEEDING S O F SYMPOSI A IN APPLIE D MATHEMATIC S Volum e 43 Combinatoria l Game s Richar d K . Guy , Edito r Elwy n R . Berlekam p Joh n Horto n Conwa y Aviezr i S . Fraenke l Richar d J. Nowakowsk i Ver a Ples s America n Mathematica l Societ y Providence , Rhod e Islan d LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE COMBINATORIAL GAMES HELD IN COLUMBUS, OHIO AUGUST 6-7, 1990 The AMS Short Course Series is sponsored by the Society's Committee on Employment and Educational Policy (CEEP). The series is under the direction of the Short Course Advisory Subcommittee of CEEP. 2000 Mathematics Subject Classification. Primary 91A05; Secondary 94B99. Library of Congress Cataloging-in-Publication Data Combinatorial games/Richard K. Guy, editor. p. cm. — (Proceedings of symposia in applied mathematics; v. 43. AMS short course lecture notes) ISBN 0-8218-0166-X 1. Game theory—Congresses. 2. Combinatorial analysis—Congresses. I. Guy, Richard K. II. Series: Proceedings of symposia in applied mathematics; v. 43. III. Series: Proceedings of symposia in applied mathematics. AMS Short course lecture notes. QA269.C65 1991 90-22771 519.3—dc20 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Bkiitorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionCmath.ams.org. Copyright © 1991 by the American Mathematical Society The American Mathematical Society retains all rights except those granted to the United States Government. ) The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Printed in the United States of America. This publication was prepared by the authors using T£jX. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 04 03 02 01 00 Table of Contents Preface xi What is a Game? RICHARD K. GUY 1 Numbers and Games JOHN HORTON CONWAY 23 Impartial Games RICHARD K. GUY 35 More Ways of Combining Games JOHN HORTON CONWAY 57 Introductory Overview of Mathematical Go Endgames ELWYN R. BERLEKAMP 73 Games and Codes VERA PLESS 101 Complexity of Games AVIEZRI S. FRAENKEL 111 ..., Welter's Game, Sylver Coinage, Dots-and-Boxes ,... RICHARD J. NOWAKOWSKI 155 Unsolved Problems in Combinatorial Games RICHARD K. GUY 183 Selected Bibliography on Combinatorial Games and Some Related Material AVIEZRI S. FRAENKEL 191 Index 227 Preface The subject of combinatorics is only slowly acquiring respectability and combinatorial games will clearly take longer than the rest of combina­ torics. Perhaps this partly stems from the puritanical view that anything amusing can't possibly involve any worthwhile mathematics. In the past, "game theory" has meant the subject delineated by von Neumann and Morgenstern, which has found wide, though usually un­ successful, application in economics, management, military strategy, and other useful forms of human activity. Combinatorial games, with com­ plete information, no chance moves, and no place for bluffing or coalitions, are of little interest to the classical game theorist, who knows that there is always at least one pure optimal strategy. Why, then, should we be interested in combinatorial games? Aviezri Praenkel gives some cogent reasons in the introduction to his Selected Bibliography at the end of this volume. There are many connexions with other parts of mathematics, only a few of which have so far begun to be explored, and only one of which is seriously considered here. Vera Pless explains the several connexions with coding theory, through which we make contact with most of the branches of the main stream of combinatorics, including graph theory. A whole new theory of number, including infinitesimals and transfinite numbers, has emerged as a special case of the theory of games. This is introduced by John Conway in the second chapter. Investigation of this remarkable area is only slowly gaining momentum, in spite of the early appearance of Donald Knuth's popularization, Surreal Numbers. Perhaps this only served to perpetuate the myth that we are dealing with a frivolous subject. As Aviezri Praenkel explains, complexity theory is very well illustrated by combinatorial games, which supply a plethora of examples of harder problems than most of those which have been considered in the past. XI xii Preface We have been able to do no more than touch on the theory of "hot" games, which are, of course, the interesting ones from a practical, as well as a theoretical point of view.
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  • When Waiting Moves You in Scoring Combinatorial Games

    When Waiting Moves You in Scoring Combinatorial Games

    WHEN WAITING MOVES YOU IN SCORING COMBINATORIAL GAMES Urban Larsson1 Dalhousie University, Canada Richard J. Nowakowski Dalhousie University, Canada Carlos P. Santos2 Center for Linear Structures and Combinatorics, Portugal Abstract Combinatorial Scoring games, with the property ‘extra pass moves for a player does no harm’, are characterized. The characterization involves an order embedding of Conway’s Normal-play games. Also, we give a theorem for comparing games with scores (numbers) which extends Ettinger’s work on dicot Scoring games. 1 Introduction The Lawyer’s offer: To settle a dispute, a court has ordered you and your oppo- nent to play a Combinatorial game, the winner (most number of points) takes all. Minutes before the contest is to begin, your opponent’s lawyer approaches you with an offer: "You, and you alone, will be allowed a pass move to use once, at any time in the game, but you must use it at some point (unless the other player runs out of moves before you used it)." Should you accept this generous offer? We will show when you should accept and when you should decline the offer. It all depends on whether Conway’s Normal-play games (last move wins) can be embedded in the ‘game’ in an order preserving way. Combinatorial games have perfect information, are played by two players who move alternately, but moreover, the games finish regardless of the order of moves. When one of the players cannot move, the winner of the game is declared by some predetermined winning condition. The two players are usually called Left (female pronoun) and Right (male pronoun).