A Family of Slow Exact k-Nim Games

Slow Exact k-Nim is a variant of the well-known game of Nim. The rules of this variant are that in each move, k of the n stacks are selected and then one token is removed from each of the k stacks. We will extend known results by proving results on the structure of the P-positions for the infinite family of Slow Exact k-Nim games where we play on all but one of the n stacks. In addition, we will introduce a more general family of "slow" Nim variants, Slow SetNim(n,A) which specifies the allowed moves via the set A. This family of games contains both Slow Exact k-Nim and the slow version of Moore's k-Nim. We give results for the infinite family of Slow SetNim(n,A) for A=-1,n\, whose P-positions are closely related to those of Slow Exact k-Nim for $k=n-1$.