The Game of Nim

Abstract

This is the binary representation of xi. By Bouton's above-mentioned condition, the numbers xi constitute a "losing combination" if and only if all the sums En= i bim are even. Since the bim are identical with the bi k, all the suns E= 1 bim are even if and only if all the sums 1 biik are even. By Bouton's condition again all the sums J=, bijk are even if and only if each of the sets of numbers aii, where j is fixed, constitutes a "losing combination." From which the conclusion follows: The numbers xi constitute a "losing combination" if and only if each of the sets aii, where j is fixed, constitutes a "losing combination." * C. L. Bouton, Annals of Math., ser. II, vol. 3, 1901, p. 35.