Nimbers are inevitable

Abstract

This article concerns the resolution of impartial combinatorial games, in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always more efficient to compute separately the nimber of at least one of the independent positions, rather than to develop directly the game tree of the sum. The concept of the nimber is therefore inevitable to accelerate the computation of impartial games, even when we only try to determine the winning or losing outcome of a starting position. We also describe algorithms to use nimbers efficiently and to conclude, we give a review of the results obtained on two impartial games: Sprouts and Cram.