International Journal of Game Theory

, Volume 47, Issue 2, pp 673–693

# Multi-player Last Nim with Passes

Original Paper

## 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 (Nns). We also present a possible explanation why determining the game values becomes more complicated if $$n\le N$$.

## Keywords

Impartial combinatorial game Multi-player Last Nim Alliance Pass

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