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Combinatorial Games, Volume 43

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AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied

Volume 43 COMBINATORIAL GAMES Edited by Richard K Guy [Columbus, Ohio, August 1990) Volume 42 CRYPTOLOGY AND COMPUTATIONAL 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

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Copyright © 1991 by the American Mathematical Society The American Mathematical Society retains all rights except those granted to the 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 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 , 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 '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. explains the significant progress that he has been making with the analysis of endgames in Go, a game long thought to be even more intractable than Chess. We introduce "impartial" games in Chapter 3. These "tepid" games are of minimal interest to the classical game theorist, but there are plenty of unsolved problems in this area, as well as in the rest of the subject. A notable one is how to deal with the "misere play" of impartial games. A small but important break-through was recently made by William Sibert, and Thane Plambeck is now unveiling the misere analysis of several games which had earlier seemed intransigent. A list of open problems is given towards the end of the book. While some of these are undoubtedly hard, inroads are being made into others even as I write, and a new generation of graduate students will find a rich vein of material waiting to be investigated. As examples of what has been and what remains to be done, Richard Nowakowski presents in the last chapter some specific examples of games. Welter's Game is now understood, but see the quotation of Berlekamp by Praenkel in section 6.2 of the "complexity" chapter. In Conway's game of Sylver Coinage, on the other hand, much remains to be discovered, though G.L. Sicherman and others are slowly revealing more and more of the truth. Finally, Berlekamp's masterly analysis of the well-known children's game of Dots-and-Boxes illustrates the many levels at which a seemingly simple game may be played, and the many quite sophisticated techniques which may be used in its analysis. We are indebted to for permission to reproduce some text and a number of figures from John Conway's and from Winning Ways for your Mathematical Plays by Berlekamp, Con• way and Guy. Thanks to the Short Course Committee for suggesting and to Jim Maxwell and Monica Foulkes for organizing the course that has supplied the raison d'etre for the present volume, and to Alison Buckser and other members of the American Mathematical Society's staff for its careful and expert production.

Richard K. Guy, The University of Calgary, 90-11-05. Index

abacus positions, 50 bogus -heap, 40 abstract game, 77 Boolean formulas, 141 acyclic, 149 Boolean functions, 152 acyclic digraph, 143 Bouton, C. L., 152 acyclic games, 129, 144 Bozulich, R., 100 addition, 32 bridge, 3 addition over, OF(2), 131 Bridg-it, 152 additivity, 81 Brussels Sprouts, 156, 182 Adleman, L., 146 bypass, 82 algorithm for, 7, 132 algorithm for, AT,P,D-labels, 121 Cannibal Games, 188 algorithm GSG, 125 canonical form, 82 algorithms, 132, 135, 145 Cantor, G., 23 algorithms of complexity, 0(n log n), 141 Captives & Suicides, 75 all-smalls, 94 capture rules, 129 All Square, 185 capturing move, 75 All the King's Horses, 64 capturing tokens, 144 annihilation, 131, 133 champion, 127 annihilation games, 127, 129, 130-131, Chandra, A. K., 141, 152 137, 143 Checkers, 113, 114, 121, 141, 143 annihilation move, 129 Chess, 3, 6, 113, 114, 116, 117, 121, 152, antonim, 185 187 arbitrary digraph, 137 chess champion, 126 arithmetically periodic games, 42 chess-like games, 129, 138, 141 arithmetic periodicity, 184 Chess on an n x n board, 141 augmented matrix, 132 chilling, 93, 94 Austin, R., 43 chomp, 150, 151, 187 Class No, 24, 32 baby ko, 94 clique, 168 Backgammon, 3 coalitions, 2 basis, 132 code digit, 42, 183 battleships, 3 code, minimum weight of, 101 Berge, C, 118 code, weight distribution of, 102 Berlekamp, E. R., 143, 149 codewords, 50 Berlekamp's Rule, 1, 9 coin-turning games, 50-54 binary code, (n, k), 101 cold game, 17, 68 binary rationals, 18 cold one, 16 binary vector, 135 colliding stars, 137 bipartite games, 144 Colon Principle, 45 birthdays, 8 common coset, 44 black holes, 137 commutative , 32 Blockbusting and , 94 complement, 78 blocking tokens, 144 complete analysis, 5 Blue-Red , 17-18, 27 complete in £xtime, 142-143 bluffing, 2 complete information, 2 board games, 152 complete problems in NP, 142

227 228 INDEX complexities of games, 141 , 32, 57 complexity, 122, 135, 149 disjunctive theory, 68 complexity classes, 141-142 distance, d(x,y) between two binary vec• complexity classification, 147 tors, 102 complexity measures, 143 distinct-size piles, 148 complexity of games, 1 dominated options, 20, 81 complexity of winning strategies, 132-133 domineering, 16, 140, 184. see also Cross- component game-graphs, 125, 149 cram composite integer, 146 Dots-and-Boxes, 2, 4, 155, 174-181, 182 compute polynomially, 131 double-cross, 174 computer simulations, 151 doublecrossed moves, 176 conditional intractability, 141 double-crossing strategy, 178 conditionally intractable games, 143, 151, Double-Dealing, 174 152 Double , 43 conditionally intractable problems, 147 downs, 95 confusion interval, 24 Downstar, 15 conjunctive, 58 Imposition, 116, 120, 125, 126, 129 conjunctive theory, 68 drawn, 3 connected finite acyclic digraph, 118 draw positions, 120-121, 130, 133 connected finite digraph, 126 draws, 7, 76, 113, 124, 128, 134, 135, 137 connected group, 75 Dudeney, 43 constant weight codes, 108-109 D.U.D.E.N.E.Y., 186 constant weight lexicodes, 109 duplicate Kayles, 43 Conway, John, 100, 126, 129, 138, 140 dynamic tie positions, 121 cooled game, 33 cooling, 93 easy strategy, 137 coset, 131 eat cakes, 184 counter function, 123, 127, 133 electrons, 137 covering radius r of a code, 102 elementary row operations, 132 crosscram, 16 empty set, 7, 26 cryptanalysis, 1 endgame, 7 cryptography, 152 ending condition, 2, 7 Curtis, Robert, 52 Enough Rope Principle, 4 cycles, 121, 122 epidemiography games, 146 cyclic digraph, 144 Epstein, R. A., 60 cyclic graphs, 123 equal, 31 equality, 25, 31 dame, 93 equality of games, 81 Davies, J., 100 equivalence classes, 59 Dawson, T. R., 43 equivalent, 11 Dawson's Chess, 43, 185 error-correcting codes, 50 Dawson's Kayles, 4, 43, 44, 180 eternal Life, 76 decimal encoding, 135 Eudoxus, 23 decision problem, 121, 144 Euler's formula, 2 decoding x, 103 Even, S., 143 Dedekind, J. W. R., 23 Even Alteration Theorem, 49, 163 Dedekind cut, 32 evil numbers, 44 Dedekind sections, 18 exhaustive search, 117 designs, 108-109 existence results, 145 deterministic machines, 127 existential problems, 142, 147 deterministic Turing Machine, 141-142 existential quantifier, 142 diamond rule, 49 exponentially large, 130-131 digital computer, 141 exponential size, 131, 149 digraphs, 5, 13, 116, 122, 127, 132, 143, extended code of C, 102 149 extended homomorphism, 132 disjoint component game-graphs, 120 £octime, 142 disjunctive, 58 disjunctive compound, 10, 115 FairGame, 155 INDEX 229 falling stars, 137 Go-Moku, 184 favorite adversary, 135 Go rules, 74 Ferguson, T. S., 129, 149 graph isomorphism, 147 Ferguson's Pairing Property, 42 graph of a game, 116-117 Fermat power, 54 Greedy Cannibals, 188 GF(2), 130 Green Hackenbush trees, 45 finite acyclic digraph, 118 Gross, Oliver, 152 finite digraphs, 125, 129, 143, 145 group of a binary code, 102 finite game-graph, 146 Grundy, P. M., 121, 149 firmament, 137 Grundy function, 51 first player win, 9 Grundy's Game, 3, 44, 183, 185 forest, 151 Guy, R. K., 100, 147, 171 forms, 11, 25, 45 Fraenkel, Aviezri S., 121, 123, 129, 140, Hackenbush strings, 18, 27, 28 141, 143, 144, 146, 151, 152, 183 Hamiltonian Cycle problem (HC), 142 freezing, 94 hardness results, 143 frieze patterns, 48-49, 161-164 HC (Hamiltonian Cycle problem), 147 Fundamental Theorem, 123, 145 hexadecimal games, 44, 184 Fundamental Theorem of Combinatorial hex on an arbitrary graph, 143 Game Theory, 128 hex on an n x n board, 143 Fusion Principle, 45, 46 high complexity, 152 fuzzy games, 12 Hilbert Nim, 157 fuzzy values, 139 Holladay, J. C, 174 homomorphisms, 131-132 9, 122 Hotcakes, 184 p-functions, 117, 123, 144, 148, 149 hot game, 16, 17, 24, 68 Gt, G cooled by t, 93 human players, 127 p-values, 147, 151 Hutchings, R. L., 169, 171 Gale, David, 150, 151, 152, 187 Gale's Vingt-et-un, 188 identical, 31 game, 7, 24, 31 identity, 31 game-graph, exponentially large, 121 identity matrix, 132 game graphs, 5-7, 79, 114-115, 116-117, immortality, 76 121, 124, 129-130, 146 impartial games, 3, 16, 35, 59, 138, 143 game on a graph, 116 incentives, 17 games and graphs, 1 incomparable pair, 11 games and numbers, 138 infinite digraphs, 144 games in which the loser wins, 145-146 infinite game-graph, 128 games with draws, 121-122 infinite ordinal numbers, 23 game trees, 5-7, 13, 79, 146 infinite ordinals, 18, 20 7-functions, 123, 144 infinite remoteness, 62 7 values, 125, 127, 131-132, 133, 135 infinitesimally close, 93 Gangolli, A., 147 infinitesimals, 18, 20, 27, 93 Gardner, M., 150, 152, 182 Infinite-Sliding Game, 157 Garey, M. R., 117, 142, 143 infinite tolls, 71 generalized Instant Insanity, 143 Infinite-Welter's Game, 165 generalized Nim-sum, 125 Innocent Marbles, 133-135 generalized Sprague-Grundy Function, input size, 147 123, 125 integer translations, 90 general strategy, 137 interstellar dust, 137 generator matrix, 101 intractability, 141, 150, 152 GF(2), 131 intractable games, 117, 141, 143, 147, 151, go, 1, 4, 17, 73, 113, 114, 121, 141, 152, 152 184 intractable problem, 117, 147 go board, 73 intractable strategy, 133 go endgames, 73-100 Ishi Press, 100 golay code, 52

Goldschmidt, E., 144 Johnson, D. S.5 117, 142, 143 230 INDEX joins, 57, 59, 62-67 Miracle Octad Generator, 52 Jones, J. P., 145 mis£re game, 166 Mis6re Kayles, 4, 55, 185 Kalmar, L., 133 Mise>e Nim, 54 Kayles, 4, 43, 44, 51, 147, 180 mis£re play, 59, 62, 67, 114, 129, 149-150 Kenyon, J. C, 43 mis£re play convention, 4, 7 Kenyon's Game, 43 mis£re remotenesses, 66 kernel, 131 Moebius transformation, 52 knapsack, 142-143 Mock Turtles, 51, 52 Ko, 76, 78, 80 Mock Turtle Theorem, 51 Kobar, 78, 80 modular Nim, 148 Kotzig, A., 183 modulus, 148 Kotzig's Nim, 148, 186 Moebius, 51, 185 kriegspiel, 3 Mogul, 51, 185 Moidores, 51 Lasker's Nim, 39, 43 Monopoly, 3 last player losing, 7 Morris, F. L., 144 last player winning, 6 moves, 2, 73 , 11, 15 move set, 148 leaves, 79 Munro, I., 143 Left and Right players, 2 left options, 35 negation, 32 Left-Right Hackenbush [ONAG, WW], 143 negative, 10, 78 left win, 9 negative game, 81 length of a path, 144 Nesetril, J., 146 Lenstra, Hendrik, 50 next move, 135, 142, 146 lexicodes, 102-108 Neyman, A., 151 Li, S.-Y.R., 140 AT-follower, 133 liberties, 75 Nim, 1, 3, 5, 10, 36, 51, 113, 114, 116, Lichtenstein, D., 141, 143 117, 147, 148, 149, 152, 156, 186 linear algebra over GF(2), 132 Nim-addition, 39, 40, 46 linear combination, 131 Nimania, 146 linear span, 132 , 36, 40 Lion and Man, 188 nimble, 37 locally path-bounded digraph, 145 nim-cube-roots, 54 locally path-finite digraph, 144 nim-heap, 35 locally trail-bounded digraph, 144 Nim-like games, 113, 116, 120-121, 122, Loebl, M., 146 129, 133, 138, 141 Long Chain Rule, 177 nim-multiplication, 53 loony position, 179 nim-pile, 114, 118 loops, 121, 122, 129, 131 nim-products, 53 loopy game, 7 nim-sequence, 41, 43 Lorberbom, M., 146 nim, strategy, 118 lose, 113 nimstring, 155, 174, 177-178 lose-win-draw outcomes, 137 nim-sums, 40, 113, 118, 119-120, 119, 131, Loyd, 43 149 Loutcome(G)' 80 nim-values, 40, 48, 51, 53, 54 Ludo, 3 no chance moves, 129 node at infinity, 76 marbles, 133, 134 non-annihilation game, 130 marking nodes, 144 nonconstructive argument, 151 markings, 91, 92 nondeterministic machines, 127 mates, 48, 159 nondeterministic Turing Machine, 142 mating method, 159-160 non-disjointness, 148, 150-151 matrix, 132 nondisjunctive move, 3 mean value, 34, 94 non-losing move, 121 (minimum excluded value), 40, 118, nonpolynomial, 117 124 normal Go endgames, 95 INDEX 231

normal play, 2, 59, 67, 114, 115, 129, 149 passes, 121 normal play convention, 6 passing, 76 Northcott's Game, 38 path-finite digraphs, 144 Norton multiplication, 95 path-finite game-graph, 144, 146 Noughts-and-Crosses, 3 path in a digraph, 144 iVPimte-complete, 142 pathological games, 144-146 ATP-completeness, 121, 142 payoff, 6 ATP-complete problem, 142 perfect information, 128 N, P, £>-labels, 121, 128 periodic games, 183 N, P, D-pattern, 115, 124, 133 periods, 43, 147 ATP-hard, 143, 144 Perl, Y., 123 NP-hardness, 123 P-follower, 133 ATP-hard problem, 142 physicists, 137 iV,P-labels, 118, 149 Plambeck, T., 147 AT-position, 114, 116, 127, 128, 133, 134, Poker, 3 144, 151 Poker Nim, 38 ^positions, 40, 60, 186 Polite Cannibals, 188 N, P-pattern, 115, 148 polynomial, 126, 143 AT,P-tool, 116 polynomial algorithms, 132, 133, 142 Number Avoidance Theorem, 17 polynomial amount of memory, 142 numbers, 17, 18, 23, 93, 95 polynomial amount of memory cells, 142 number tree, 28 polynomial amount of time, 142 n x m matrix, 150 polynomial construction, 123 n x n boards, 143, 152 polynomial games, 152 n x n chess board, 141 poly normality, 150 polynomial solution, 141, 151 polynomial strategy, 137, 141, 147, 151, octal games, 42, 147, 183 152 odious numbers, 44, 51, 52 polynomial strategy in the broad sense, officers, 43, 183, 185 133 ONAG, 123-124, 147 polynomial strategy of play, 133 One for Left, Two for Right, Free for All, polynomial time, 120 69 polynomial time strategy, 125 On Numbers and Games, 155, 159, 163 poset games, 150 ^-positions, 60, 62, 186 position, 6 opposite game, 32 positive game, 81 opposite type, 137 positrons, 137 optimal next move, 133 Positrons & Electrons, 188 optimal strategy, 128 possible cycling, 133 options, 2, 7 ^-position, 40, 52, 60, 186 orbits, 137 P-position, 114, 116, 125, 126, 129-130, order, 31, 32 144 ordinal numbers, 8, 36 preperiods, 147 ordinal sum, 46 primes of the form n, s8, +2, =, slO-f-1, ordinary Nim-sum, 124 145 ordinary sum, 129 primes, 146 outcome classes, 9 Princess and the Roses, The, 187 outcomes, 6, 59, 80 probabilistic method, 146 overheating, 4, 17, 94 product, 117 overriding move, 71 professional chess-player, 126 proper sum, 58 pairing strategy, 167 provably and conditionally intractable Parity Rule, 174, 176 games, 141-146 partially ordered sets, 150 provably intractable games, 141 partial ordering of games, 81 pseudonumber, 24 Particles and Antiparticles, 135-137 Pspace, 142, 143 partizan games, 3, 16, 38, 59, 69, 138-140, Pspace-complete, 143 144 Pspace-complete games, 148 232 INDEX

Pspace-completeness, 144 simplifying games, 20 Pspace-complete problem, 142 simultaneous plays, 57-59 Pspace-hard, 142, 143, 144 Sipser, M., 143 Put-or-Take-a-Square, 60-61, 186 Sliding Game, 156-157, 182 slow joins, 58, 63, 65, 68 quadratic residue codes, 52, 105-108 Smith, C.A.B., 59, 123, 133, 147, 149 quick sum, 58 Snakes-and-Ladders, 3 Quiet End Theorem, 170 solitaire game, 143 quotient spaces, 131-132 Solovay, R., 146 sorting, 147 Rabin, M. O., 145, 146 sorting n numbers, 141 rapid joins, 58, 63, 95 space, 131 rare values, 44 space stations, 135 real-closed Field, 25 sparse space, 44 real numbers, 27 spinster, 48 rectangle inequality, 26 Sprague, R., 121 Redwood Furniture, 143 Sprague-Grundy Function, 117-120 Reisch, S., 143 Sprague-Grundy theorem, 35 remoteness, 58, 61, 63, 65, 67 Sprague-Grundy theory, 54, 121, 172 remoteness function, 59-62 Sprouts, 187 remoteness, mis£re, 66 squares Off, 184 standard input size, 120, 125 removing nodes, 144 standard size, 120 repetitious positions, 76 stars, 9, 95, 135 reversible moves, 40, 82, 157 Steiner system, 52 reversible options, 20, 88 Steinhaus, H., 59 right options, 35 Stockmeyer, L. J., 141, 152 right win, 9 stopping position, 100 Rivest, R. L., 146 Strassen, V., 146 Robertson, E., 143 strategy, 117, 122, 123, 129, 133, 137, 141, Robson, J. M., 141 147 robustness, 146 strategy in the narrow sense, 133 row-echelon form, 132 strategy of play, 132 RSA public-key cryptosystem, 146 strategy in the wide sense, 133 Routcome(^)? 80 strategy-stealing, 169 Ruler Game, 52 Strings-and-Coins, 177-178 strong Murphy Law, 121-123 Saltus «, 42, 43 subClass, 33 Schaefer, T. J., 143, 148 subset Take-away, 188 Scheinerman, E. R., 151 subspaces, 131 Schuhstrings, 186 subtraction games, 41, 183 Scissors-Paper-Stone, 3 subtraction set, 41, 183 score, 6 succinct games, 147-148 second player win, 9 succinctness, 120, 150-151 selective, 58 succinctness of input size, 148 selective game, 69 suicidal move, 71, 75 selective theory, 68 sum, 7, 10, 116-117, 121, 124, 140, 147, set of options, 11 149, 150 sets of options, 6 sum-game, 129 Shaki, A. S., 140 sum-graph, 117, 121 Shamir, A., 146 sum of games, 77, 115, 140 She-Loves-Me-She-Loves-Me-Not, 41, 182 superset game, 151 Sicherman, G. L., 172 superspacemen, 137 Silver Dollar Game, 37 Surreal Numbers, 24 simpler games, 8 suspense numbers, 58, 63, 65, 66, 69 simplest number, 69 suspension bridge, 80 simplicity, 8 Sylver Coinage, 4, 155, 166-174, 182, 187 Simplicity Theorem, 140 Sylvester, J. J., 166 INDEX 233 take-and-break games, 42-45 twopins theory, 175 tally, 68 twopins-vine, 180 Tally (Toll & Timer), 58 tardy unions, 58, 71 Ulehla, J., 151 Tarjan, R. E., 143 ultimately periodic games, 42 Tartan Theorem, 53 undecidability results, 145 Tassa, U., 129, 140 undecidable game, 145 tax, 95 unions, 57, 68 temperature, 34 unit vectors, 131 tepid game, 68, 69 Up, 15 terminal nodes, 79 upper bound, 132 terminal position, 6 ups, 95 ternary expansion, 52 Upstar, 15 Ternups, 52 urgent unions, 58, 71 thermograph, 33 Thomas, Hugh, 157 values, 7, 11, 45 Three-Finger Morra, 3 vector addition over GF(2), 113 Tic-TaoToe, 3 vector spaces, 131 tie, 6, 7, 76 Voelkle, F., 171 tied, 3 von Neumann, John, 24, 36 timer, 69 von Neumann's game, 187 tinies, 95 von Neumann's Hackendot, 151 toll, 68 Top Entails, 185 warming, 93, 94 totally ordered Field, 32 weight of a vector, 101 totally ordered numbers, 25, 26, 33 Welter, 137, 148 tractability, 141 Welter function, 48, 49 tractable games, 117, 120, 143-144 Welter's Game, 47-50, 155, 158-159, 182, tractable problems, 147 188 tractable strategy, 133 Welter's Nim, 149 trail, 144 win, 113 trail-finite, 144 winning move, 128, 135, 151 transitivity, 81 winning strategy, 5 Traveling Salesman problem (TS), 1, 142 Winning Ways, 155, 171, 179, 180 treblecross, 43, 69, 183 Wonderland, 147 tree, 142, 151 Worlds, 135 tree of numbers, 30 Worlds in Collision, 135-137 tribulations, 186 WW, 123, 147 triple Ko, 76 Wythoff's Game, 3, 156 triplicate Nim, 43 TS (Traveling Salesman), 147 Yamasaki, Y., 149 Turning Turtles, 51 Yesha, Y., 123, 129, 143, 144 Turnips, 52 Tweedledum and Tweedledee Principle, 36 zero, 26 2-person game, 128 zero-one game, 113 2-player full information games, 113 , defined, 11 twopins, 180-181

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