Multi-player Last Nim with Passes

Abstract

We introduce a class of impartial combinatorial games, Multi-player Last Nim with Passes, denoted by MLNim$^{(s)}(N,n)$: there are N piles of counters which are linearly ordered. In turn, each of n players either removes any positive integer of counters from the last pile, or makes a choice ‘pass’. Once a ‘pass’ option is used, the total number s of passes decreases by 1. When all s passes are used, no player may ever ‘pass’ again. A pass option can be used at any time, up to the penultimate move, but cannot be used at the end of the game. The player who cannot make a move wins the game. The aim is to determine the game values of the positions of MLNim$^{(s)}(N,n)$ for all integers $N\ge 1$ and $n\ge 3$ and $s\ge 1$. For $n>N+1$ or $n=N+1\ge 3$, the game values are completely determined for any $s\ge 1$. For $3\le n\le N$, the game values are determined for infinitely many triplets (N, n, s). We also present a possible explanation why determining the game values becomes more complicated if $n\le N$.