ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

On numbers and games, by J. H. Conway. Pp ix, 238. £6·50. 1976. SBN 0 12 186350 6 (Academic Press)

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive

http://ieeexplore.ieee.org/iel5/5/31266/01455274.pdf

On numbers and games

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is go. The shorter terminology game, games is used below to designate combinatorial games.

Combinatorial games: Selected bibliography with a succinct gourmet introduction.

We have retained the numbering from the list of unsolved problems given on pp. 183–189 of Amer. Math. Soc. Proc. Sympos. Appl Math. 43(1991) and added in some new material. For many more references than we list, see Fraenkel’s Bibliography.

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Unsolved Problems in Combinatorial Games

We introduce a misere quotient semigroup construction in impartial combinatorial game theory, and argue that it is the long-sought natural generalization of the normal-play Sprague-Grundy theory to misere play. Along the way, we illustrate how to use the theory to describe complete analyses of two wild taking and breaking games.

/pdf/taming-the-wild-in-impartial-combinatorial-games-rgzqt5h4df.pdf

Taming the wild in impartial combinatorial games

Misère quotients for impartial games

The nim-sequences of the octal games ·127, ·16, ·376, and ·56 are periodic. The results are obtained using Berlekamp's method of sparse spaces and common cosets. We also extend the range of known values for the nim-sequences of fifty-four other octal games.

Thane E. Plambeck

Papers

Taming the wild in impartial combinatorial games

Misère quotients for impartial games

A note on periodicity in some octal games

Daisies, Kayles, and the Sibert-Conway decomposition in misegre octal games

New infinite families of simple 5-designs