Winning Positions in Simplicial Nim

Abstract

Simplicial Nim, introduced by Ehrenborg and Steingrímsson, is a generalization of the classical two-player game of Nim. The heaps are placed on the vertices of a simplicial complex and a player's move may affect any number of piles provided that the corresponding vertices form a face of the complex. In this paper, we present properties of a complex that are equivalent to the $\cal P$-positions (winning positions for the second player) being closed under addition. We provide examples of such complexes and answer a number of open questions posed by Ehrenborg and Steingrímsson.