Multi-player Last Nim with Passes

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 \(n\le N\).