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Published online by Cambridge University Press: 27 June 2025
This periodically updated reference resource is intended to put eager researchers on the path to fame and (perhaps) fortune.
As in our earlier articles, WW stands for Winning Ways [Berlekamp et al. 1982]. We say that the nim-value of a position is n when its value is the nimber *n.
1. Subtraction games are known to be periodic. Investigate the relationship between the subtraction set and the length and structure of the period.
(For subtraction games, see WW, pp. 83-86, 487-498, and Section 4 of [Guy 1996] in this volume, especially the table on page 67. A move in the subtraction 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 in this particular game is: in symbols, Arbitrarily difficult subtraction games can be devised by using infinite subtraction sets: the primes, for example.) The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set [Fraenkel and Kotzig 1987].
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