Playing Nim on a Simplicial Complex

Abstract

We introduce a generalization of the classical game of Nim by placing the piles on the vertices of a simplicial complex and allowing a move to affect the piles on any set of vertices that forms a face of the complex. Under certain conditions on the complex we present a winning strategy. These conditions are satisfied, for instance, when the simplicial complex consists of the independent sets of a binary matroid. Moreover, we study four operations on a simplicial complex under which games on the complex behave nicely. We also consider particular complexes that correspond to natural generalizations of classical Nim.