Slow $k$-Nim

Abstract

Given $n$ piles of tokens and a positive integer $k\le n$, we study the following two impartial combinatorial games Nim${}_{n,\le k}^1$ and Nim${}_{n,=k}^1$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and misère version of each game we compute the Sprague-Grundy function for the cases $n=k=2$ and $n=k+1=3$. For game Nim${}_{n,\le k}^1$ we also characterize its P-positions for the cases $n\le k+2$ and $n=k+3\le 6$.