On the Sprague-Grundy function of Exact $k$-Nim

Abstract

Moore's generalization of the game of {\sc Nim} is played as follows. Let $n$ and $k$ be two integers such that $1 \leq k \leq n$. Given $n$ piles of tokens, two players move alternately, removing tokens from at least one and at most $k$ of the piles. The player who makes the last move wins. The game was solved by Moore in 1910 and an explicit formula for its Sprague-Grundy function was given by Jenkyns and Mayberry in 1980, for the case $n = k+1$ only. We introduce another generalization of {\sc Nim}, called {\sc Exact $k$-Nim}, in which each move reduces exactly $k$ piles. We give an explicit formula for the Sprague-Grundy function of {\sc Exact $k$-Nim} in case $2k \geq n$. In case $n=2k$ our formula is surprisingly similar to Jenkyns and Mayberry's one.