Unsolved Problems in Combinatorial Game Theory: Updated

Summary

We have retained the numbering from the list of unsolved problems given on pp. 183-189 of AMS Proc. Sympos. Appl. Math. 43(1991), called PS AM 43 below, and on pp. 475-491 of this volume's predecessor, Games of No Chance, hereafter referred to as GONC. This list also contains more detail about some of the games mentioned below. References in brackets, e.g., Ferguson [1974], are listed in Fraenkel's Bibliography later in this book; WW refers to

Elwyn Berlekamp, John Conway and Richard Guy, Winning Ways for your Mathematical Plays, Academic Press, 1982. A.K.Peters, 2000.

and references in parentheses, e.g., Kraitchik (1941), are at the end of this article.

1. Subtraction games are known to be periodic. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set. See Fraenkel and Kotzig [1987].

See also Subtraction Games in WW, 83-86, 487-498 and in the Impartial Games article in GONC. A move in the game S(s₁, s₂, s₃,…) is to take a number of beans from a heap, provided that number is a member of the subtraction-set, {s₁, s₂, s₃,…}. Analysis of such a game and of many other heap games is conveniently recorded by a nim-sequence,

meaning that the nim-value of a heap of h beans is n_h, h = 0, 1, 2, …, i.e., that the value of a heap of h beans in this particular game is the nimbern_h. To avoid having to print stars, we say that the nim-value of a position is n, meaning that its value is the nimber *n.