Multi-player End-Nim games

Abstract

W.O. Krawec (2012) [17] introduced a method of analyzing multi-player impartial games, and derived a recursive function capable of determining which of the n players has a winning strategy. The present paper is devoted to “Multi-player End-Nim games”, abbreviated by ENim($N,n$), assuming that the standard alliance matrix is adopted. The game values of ENim($N,n$) are completely determined for three cases $n>N+1$, $n=N+1$ and $n=N$. For the case $n<N$, by letting $d=N-n\ge 1$, the game values of ENim($n+d,n$) are completely determined if $n\ge d+4$. The case $3\le n\le d+3$ is more complicated, we present the game values for $d=1$.