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AMS SHORT COURSE LECTURE NOTES Introductor y Surve y Lecture s

publishe d as 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.

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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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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 combina­torics. 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 un­successful, application in economics, management, military strategy, and other useful forms of human activity. Combinatorial games, with com­plete 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

xi i 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. Elwyn Berlekamp 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 Academic Press for permission to reproduce some text and a number of figures from John Conway's On Numbers and Games and from Winning Ways for your Mathematical Plays by Berlekamp, Con­way 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 abstract game, 77 acyclic, 149 acyclic digraph, 143 acyclic games, 129, 144 addition, 32 addition over, OF(2), 131 additivity, 81 Adleman, L., 146 algorithm for, 7, 132 algorithm for, AT,P,D-labels, 121 algorithm GSG, 125 algorithms, 132, 135, 145 algorithms of complexity, 0(n log n), 141 all-smalls, 94 All Square, 185 All the King's Horses, 64 annihilation, 131, 133 annihilation games, 127, 129, 130-131,

137, 143 annihilation move, 129 antonim, 185 arbitrary digraph, 137 arithmetically periodic games, 42 arithmetic periodicity, 184 augmented matrix, 132 Austin, R., 43

baby ko, 94 Backgammon, 3 basis, 132 battleships, 3 Berge, C , 118 Berlekamp, E. R., 143, 149 Berlekamp's Rule, 1, 9 binary code, (n, k), 101 binary rationals, 18 binary vector, 135 bipartite games, 144 birthdays, 8 black holes, 137 Blockbusting and Domineering, 94 blocking tokens, 144 Blue-Red Hackenbush, 17-18, 27 bluffing, 2 board games, 152

bogus nim-heap, 40 Boolean formulas, 141 Boolean functions, 152 Bouton, C. L., 152 Bozulich, R., 100 bridge, 3 Bridg-it, 152 Brussels Sprouts, 156, 182 bypass, 82

Cannibal Games, 188 canonical form, 82 Cantor, G., 23 Captives & Suicides, 75 capture rules, 129 capturing move, 75 capturing tokens, 144 champion, 127 Chandra, A. K., 141, 152 Checkers, 113, 114, 121, 141, 143 Chess, 3, 6, 113, 114, 116, 117, 121, 152,

187 chess champion, 126 chess-like games, 129, 138, 141 Chess on an n x n board, 141 chilling, 93, 94 chomp, 150, 151, 187 Class No, 24, 32 clique, 168 coalitions, 2 code digit, 42, 183 code, minimum weight of, 101 code, weight distribution of, 102 codewords, 50 coin-turning games, 50-54 cold game, 17, 68 cold one, 16 colliding stars, 137 Colon Principle, 45 common coset, 44 commutative group, 32 complement, 78 complete analysis, 5 complete in £xt ime, 142-143 complete information, 2 complete problems in NP, 142

227

228

complexities of games, 141 complexity, 122, 135, 149 complexity classes, 141-142 complexity classification, 147 complexity measures, 143 complexity of games, 1 complexity of winning strategies, 132-133 component game-graphs, 125, 149 composite integer, 146 compute polynomially, 131 computer simulations, 151 conditional intractability, 141 conditionally intractable games, 143, 151,

152 conditionally intractable problems, 147 confusion interval, 24 conjunctive, 58 conjunctive theory, 68 connected finite acyclic digraph, 118 connected finite digraph, 126 connected group, 75 constant weight codes, 108-109 constant weight lexicodes, 109 Conway, John, 100, 126, 129, 138, 140 cooled game, 33 cooling, 93 coset, 131 counter function, 123, 127, 133 covering radius r of a code, 102 crosscram, 16 cryptanalysis, 1 cryptography, 152 Curtis, Robert, 52 cycles, 121, 122 cyclic digraph, 144 cyclic graphs, 123

dame, 93 Davies, J., 100 Dawson, T. R., 43 Dawson's Chess, 43, 185 Dawson's Kayles, 4, 43, 44, 180 decimal encoding, 135 decision problem, 121, 144 decoding x, 103 Dedekind, J. W. R., 23 Dedekind cut, 32 Dedekind sections, 18 designs, 108-109 deterministic machines, 127 deterministic Turing Machine, 141-142 diamond rule, 49 digital computer, 141 digraphs, 5, 13, 116, 122, 127, 132, 143,

149 disjoint component game-graphs, 120 disjunctive, 58 disjunctive compound, 10, 115

INDEX

disjunctive sum, 32, 57 disjunctive theory, 68 distance, d(x,y) between two binary vec­

tors, 102 distinct-size piles, 148 dominated options, 20, 81 domineering, 16, 140, 184. see also Cross-

cram Dots-and-Boxes, 2, 4, 155, 174-181, 182 double-cross, 174 doublecrossed moves, 176 double-crossing strategy, 178 Double-Dealing, 174 Double Kayles, 43 downs, 95 Downstar, 15 Imposition, 116, 120, 125, 126, 129 drawn, 3 draw positions, 120-121, 130, 133 draws, 7, 76, 113, 124, 128, 134, 135, 137 Dudeney, 43 D.U.D.E.N.E.Y., 186 duplicate Kayles, 43 dynamic tie positions, 121

easy strategy, 137 eat cakes, 184 electrons, 137 elementary row operations, 132 empty set, 7, 26 endgame, 7 ending condition, 2, 7 Enough Rope Principle, 4 epidemiography games, 146 Epstein, R. A., 60 equal, 31 equality, 25, 31 equality of games, 81 equivalence classes, 59 equivalent, 11 error-correcting codes, 50 eternal Life, 76 Eudoxus, 23 Euler's formula, 2 Even, S., 143 Even Alteration Theorem, 49, 163 evil numbers, 44 exhaustive search, 117 existence results, 145 existential problems, 142, 147 existential quantifier, 142 exponentially large, 130-131 exponential size, 131, 149 extended code of C, 102 extended homomorphism, 132 £octime, 142

FairGame, 155

INDEX 229

falling stars, 137 favorite adversary, 135 Ferguson, T. S., 129, 149 Ferguson's Pairing Property, 42 Fermat power, 54 field GF(2), 130 finite acyclic digraph, 118 finite digraphs, 125, 129, 143, 145 finite game-graph, 146 firmament, 137 first player win, 9 forest, 151 forms, 11, 25, 45 Fraenkel, Aviezri S., 121, 123, 129, 140,

141, 143, 144, 146, 151, 152, 183 freezing, 94 frieze patterns, 48-49, 161-164 Fundamental Theorem, 123, 145 Fundamental Theorem of Combinatorial

Game Theory, 128 Fusion Principle, 45, 46 fuzzy games, 12 fuzzy values, 139

9, 122 p-functions, 117, 123, 144, 148, 149 Gt, G cooled by t, 93 p-values, 147, 151 Gale, David, 150, 151, 152, 187 Gale's Vingt-et-un, 188 game, 7, 24, 31 game-graph, exponentially large, 121 game graphs, 5-7, 79, 114-115, 116-117,

121, 124, 129-130, 146 game on a graph, 116 games and graphs, 1 games and numbers, 138 games in which the loser wins, 145-146 games with draws, 121-122 game trees, 5-7, 13, 79, 146 7-functions, 123, 144 7 values, 125, 127, 131-132, 133, 135 Gangolli, A., 147 Gardner, M., 150, 152, 182 Garey, M. R., 117, 142, 143 generalized Instant Insanity, 143 generalized Nim-sum, 125 generalized Sprague-Grundy Function,

123, 125 general strategy, 137 generator matrix, 101 GF(2), 131 go, 1, 4, 17, 73, 113, 114, 121, 141, 152,

184 go board, 73 go endgames, 73-100 golay code, 52 Goldschmidt, E., 144

Go-Moku, 184 Go rules, 74 graph isomorphism, 147 graph of a game, 116-117 Greedy Cannibals, 188 Green Hackenbush trees, 45 Gross, Oliver, 152 group of a binary code, 102 Grundy, P. M., 121, 149 Grundy function, 51 Grundy's Game, 3, 44, 183, 185 Guy, R. K., 100, 147, 171

Hackenbush strings, 18, 27, 28 Hamiltonian Cycle problem (HC), 142 hardness results, 143 HC (Hamiltonian Cycle problem), 147 hexadecimal games, 44, 184 hex on an arbitrary graph, 143 hex on an n x n board, 143 high complexity, 152 Hilbert Nim, 157 Holladay, J. C , 174 homomorphisms, 131-132 Hotcakes, 184 hot game, 16, 17, 24, 68 human players, 127 Hutchings, R. L., 169, 171

identical, 31 identity, 31 identity matrix, 132 immortality, 76 impartial games, 3, 16, 35, 59, 138, 143 incentives, 17 incomparable pair, 11 infinite digraphs, 144 infinite game-graph, 128 infinite ordinal numbers, 23 infinite ordinals, 18, 20 infinite remoteness, 62 infinitesimally close, 93 infinitesimals, 18, 20, 27, 93 Infinite-Sliding Game, 157 infinite tolls, 71 Infinite-Welter's Game, 165 Innocent Marbles, 133-135 input size, 147 integer translations, 90 interstellar dust, 137 intractability, 141, 150, 152 intractable games, 117, 141, 143, 147, 151,

152 intractable problem, 117, 147 intractable strategy, 133 Ishi Press, 100

Johnson, D. S.5 117, 142, 143

230 INDEX

joins, 57, 59, 62-67 Jones, J. P., 145

Kalmar, L., 133 Kayles, 4, 43, 44, 51, 147, 180 Kenyon, J. C , 43 Kenyon's Game, 43 kernel, 131 knapsack, 142-143 Ko, 76, 78, 80 Kobar, 78, 80 Kotzig, A., 183 Kotzig's Nim, 148, 186 kriegspiel, 3

Lasker's Nim, 39, 43 last player losing, 7 last player winning, 6 lattice, 11, 15 leaves, 79 Left and Right players, 2 left options, 35 Left-Right Hackenbush [ONAG, WW], 143 left win, 9 length of a path, 144 Lenstra, Hendrik, 50 lexicodes, 102-108 Li, S.-Y.R., 140 liberties, 75 Lichtenstein, D., 141, 143 linear algebra over GF(2), 132 linear combination, 131 linear span, 132 Lion and Man, 188 locally path-bounded digraph, 145 locally path-finite digraph, 144 locally trail-bounded digraph, 144 Loebl, M., 146 Long Chain Rule, 177 loony position, 179 loops, 121, 122, 129, 131 loopy game, 7 Lorberbom, M., 146 lose, 113 lose-win-draw outcomes, 137 Loyd, 43 Loutcome(G)' 8 0

Ludo, 3

marbles, 133, 134 marking nodes, 144 markings, 91, 92 mates, 48, 159 mating method, 159-160 matrix, 132 mean value, 34, 94 mex (minimum excluded value), 40, 118,

124

Miracle Octad Generator, 52 mis£re game, 166 Mis6re Kayles, 4, 55, 185 Mise>e Nim, 54 mis£re play, 59, 62, 67, 114, 129, 149-150 mis£re play convention, 4, 7 mis£re remotenesses, 66 Moebius transformation, 52 Mock Turtles, 51, 52 Mock Turtle Theorem, 51 modular Nim, 148 modulus, 148 Moebius, 51, 185 Mogul, 51, 185 Moidores, 51 Monopoly, 3 Morris, F. L., 144 moves, 2, 73 move set, 148 Munro, I., 143

negation, 32 negative, 10, 78 negative game, 81 Nesetril, J., 146 next move, 135, 142, 146 Neyman, A., 151 AT-follower, 133 Nim, 1, 3, 5, 10, 36, 51, 113, 114, 116,

117, 147, 148, 149, 152, 156, 186 Nim-addition, 39, 40, 46 Nimania, 146 nimbers, 36, 40 nimble, 37 nim-cube-roots, 54 nim-heap, 35 Nim-like games, 113, 116, 120-121, 122,

129, 133, 138, 141 nim-multiplication, 53 nim-pile, 114, 118 nim-products, 53 nim-sequence, 41, 43 nim, strategy, 118 nimstring, 155, 174, 177-178 nim-sums, 40, 113, 118, 119-120, 119, 131,

149 nim-values, 40, 48, 51, 53, 54 no chance moves, 129 node at infinity, 76 non-annihilation game, 130 nonconstructive argument, 151 nondeterministic machines, 127 nondeterministic Turing Machine, 142 non-disjointness, 148, 150-151 nondisjunctive move, 3 non-losing move, 121 nonpolynomial, 117 normal Go endgames, 95

INDEX 231

normal play, 2, 59, 67, 114, 115, 129, 149 normal play convention, 6 Northcott's Game, 38 Norton multiplication, 95 Noughts-and-Crosses, 3 iVPimte-complete, 142 ATP-completeness, 121, 142 ATP-complete problem, 142 N, P, £>-labels, 121, 128 N, P, D-pattern, 115, 124, 133 ATP-hard, 143, 144 NP-hardness, 123 ATP-hard problem, 142 iV,P-labels, 118, 149 AT-position, 114, 116, 127, 128, 133, 134,

144, 151 ^positions, 40, 60, 186 N, P-pattern, 115, 148 AT,P-tool, 116 Number Avoidance Theorem, 17 numbers, 17, 18, 23, 93, 95 number tree, 28 n x m matrix, 150 n x n boards, 143, 152 n x n chess board, 141

octal games, 42, 147, 183 odious numbers, 44, 51, 52 officers, 43, 183, 185 ONAG, 123-124, 147 One for Left, Two for Right, Free for All,

69 On Numbers and Games, 155, 159, 163 ^-positions, 60, 62, 186 opposite game, 32 opposite type, 137 optimal next move, 133 optimal strategy, 128 options, 2, 7 orbits, 137 order, 31, 32 ordinal numbers, 8, 36 ordinal sum, 46 ordinary Nim-sum, 124 ordinary sum, 129 outcome classes, 9 outcomes, 6, 59, 80 overheating, 4, 17, 94 overriding move, 71

pairing strategy, 167 Parity Rule, 174, 176 partially ordered sets, 150 partial ordering of games, 81 Particles and Antiparticles, 135-137 partizan games, 3, 16, 38, 59, 69, 138-140,

144

passes, 121 passing, 76 path-finite digraphs, 144 path-finite game-graph, 144, 146 path in a digraph, 144 pathological games, 144-146 payoff, 6 perfect information, 128 periodic games, 183 periods, 43, 147 Perl, Y., 123 P-follower, 133 physicists, 137 Plambeck, T., 147 Poker, 3 Poker Nim, 38 Polite Cannibals, 188 polynomial, 126, 143 polynomial algorithms, 132, 133, 142 polynomial amount of memory, 142 polynomial amount of memory cells, 142 polynomial amount of time, 142 polynomial construction, 123 polynomial games, 152 poly normality, 150 polynomial solution, 141, 151 polynomial strategy, 137, 141, 147, 151,

152 polynomial strategy in the broad sense,

133 polynomial strategy of play, 133 polynomial time, 120 polynomial time strategy, 125 poset games, 150 position, 6 positive game, 81 positrons, 137 Positrons & Electrons, 188 possible cycling, 133 ^-position, 40, 52, 60, 186 P-position, 114, 116, 125, 126, 129-130,

144 preperiods, 147 primes of the form n, s8, +2, =, slO-f-1,

145 primes, 146 Princess and the Roses, The, 187 probabilistic method, 146 product, 117 professional chess-player, 126 proper sum, 58 provably and conditionally intractable

games, 141-146 provably intractable games, 141 pseudonumber, 24 Pspace, 142, 143 Pspace-complete, 143 Pspace-complete games, 148

INDEX 232

Pspace-completeness, 144 Pspace-complete problem, 142 Pspace-hard, 142, 143, 144 Put-or-Take-a-Square, 60-61, 186

quadratic residue codes, 52, 105-108 quick sum, 58 Quiet End Theorem, 170 quotient spaces, 131-132

Rabin, M. O., 145, 146 rapid joins, 58, 63, 95 rare values, 44 real-closed Field, 25 real numbers, 27 rectangle inequality, 26 Redwood Furniture, 143 Reisch, S., 143 remoteness, 58, 61, 63, 65, 67 remoteness function, 59-62 remoteness, mis£re, 66

removing nodes, 144 repetitious positions, 76 reversible moves, 40, 82, 157 reversible options, 20, 88 right options, 35 right win, 9 Rivest, R. L., 146 Robertson, E., 143 Robson, J. M., 141 robustness, 146 row-echelon form, 132 RSA public-key cryptosystem, 146

Routcome(^)? 80 Ruler Game, 52

Saltus «, 42, 43 Schaefer, T. J., 143, 148 Scheinerman, E. R., 151 Schuhstrings, 186 Scissors-Paper-Stone, 3 score, 6 second player win, 9 selective, 58 selective game, 69 selective theory, 68 set of options, 11 sets of options, 6 Shaki, A. S., 140 Shamir, A., 146 She-Loves-Me-She-Loves-Me-Not, 41, 182 Sicherman, G. L., 172 Silver Dollar Game, 37 simpler games, 8 simplest number, 69 simplicity, 8 Simplicity Theorem, 140

simplifying games, 20 simultaneous plays, 57-59 Sipser, M., 143 Sliding Game, 156-157, 182 slow joins, 58, 63, 65, 68 Smith, C.A.B., 59, 123, 133, 147, 149 Snakes-and-Ladders, 3 solitaire game, 143 Solovay, R., 146 sorting, 147 sorting n numbers, 141 space, 131 space stations, 135 sparse space, 44 spinster, 48 Sprague, R., 121 Sprague-Grundy Function, 117-120 Sprague-Grundy theorem, 35 Sprague-Grundy theory, 54, 121, 172 Sprouts, 187 squares Off, 184 standard input size, 120, 125 standard size, 120 stars, 9, 95, 135 Steiner system, 52 Steinhaus, H., 59 Stockmeyer, L. J., 141, 152 stopping position, 100 Strassen, V., 146 strategy, 117, 122, 123, 129, 133, 137, 141,

147 strategy in the narrow sense, 133 strategy of play, 132 strategy in the wide sense, 133 strategy-stealing, 169 Strings-and-Coins, 177-178 strong Murphy Law, 121-123 subClass, 33 subset Take-away, 188 subspaces, 131 subtraction games, 41, 183 subtraction set, 41, 183 succinct games, 147-148 succinctness, 120, 150-151 succinctness of input size, 148 suicidal move, 71, 75 sum, 7, 10, 116-117, 121, 124, 140, 147,

149, 150 sum-game, 129 sum-graph, 117, 121 sum of games, 77, 115, 140 superset game, 151 superspacemen, 137 Surreal Numbers, 24 suspense numbers, 58, 63, 65, 66, 69 suspension bridge, 80 Sylver Coinage, 4, 155, 166-174, 182, 187 Sylvester, J. J., 166

INDEX

take-and-break games, 42-45 tally, 68 Tally (Toll & Timer), 58 tardy unions, 58, 71 Tarjan, R. E., 143 Tartan Theorem, 53 Tassa, U., 129, 140 tax, 95 temperature, 34 tepid game, 68, 69 terminal nodes, 79 terminal position, 6 ternary expansion, 52 Ternups, 52 thermograph, 33 Thomas, Hugh, 157 Three-Finger Morra, 3 Tic-TaoToe, 3 tie, 6, 7, 76 tied, 3 timer, 69 tinies, 95 toll, 68 Top Entails, 185 totally ordered Field, 32 totally ordered numbers, 25, 26, 33 tractability, 141 tractable games, 117, 120, 143-144 tractable problems, 147 tractable strategy, 133 trail, 144 trail-finite, 144 transitivity, 81 Traveling Salesman problem (TS), 1, 142 treblecross, 43, 69, 183 tree, 142, 151 tree of numbers, 30 tribulations, 186 triple Ko, 76 triplicate Nim, 43 TS (Traveling Salesman), 147 Turning Turtles, 51 Turnips, 52 Tweedledum and Tweedledee Principle, 36 2-person game, 128 2-player full information games, 113 twopins, 180-181

233

twopins theory, 175 twopins-vine, 180

Ulehla, J., 151 ultimately periodic games, 42 undecidability results, 145 undecidable game, 145 unions, 57, 68 unit vectors, 131 Up, 15 upper bound, 132 ups, 95 Upstar, 15 urgent unions, 58, 71

values, 7, 11, 45 vector addition over GF(2), 113 vector spaces, 131 Voelkle, F., 171 von Neumann, John, 24, 36 von Neumann's game, 187 von Neumann's Hackendot, 151

warming, 93, 94 weight of a vector, 101 Welter, 137, 148 Welter function, 48, 49 Welter's Game, 47-50, 155, 158-159, 182,

188 Welter's Nim, 149 win, 113 winning move, 128, 135, 151 winning strategy, 5 Winning Ways, 155, 171, 179, 180 Wonderland, 147 Worlds, 135 Worlds in Collision, 135-137 WW, 123, 147 Wythoff's Game, 3, 156

Yamasaki, Y., 149 Yesha, Y., 123, 129, 143, 144

zero, 26 zero-one game, 113 zero game, defined, 11