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. 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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. 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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.