Nimbers are inevitable

TL;DR

This paper demonstrates that calculating nimbers separately for independent game positions is more efficient than analyzing the combined game directly, emphasizing the fundamental role of nimbers in solving impartial combinatorial games.

Contribution

It proves the inevitability of nimbers for efficiently determining outcomes in impartial games and introduces algorithms for their effective computation.

Findings

Calculating nimbers separately is more efficient than direct game tree analysis.

Nimbers are essential for solving impartial games.

Algorithms for nimber computation are presented.

Abstract

This article concerns the resolution of impartial combinatorial games, and 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 nimbers of each independent position than to develop directly the game tree of the sum. The concept of nimber is therefore inevitable to solve impartial games, even when we only try to determinate the winning or losing outcome of a starting position. We also describe algorithms to use nimbers efficiently and finally, we give a review of the results obtained on two impartial games: Sprouts and Cram.