International Journal of Game Theory

, Volume 47, Issue 2, pp 673–693

# Multi-player Last Nim with Passes

• Wen An Liu
• Juan Yang
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

## Notes

### Acknowledgements

The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have greatly improved the earlier version of this paper. The research is supported by the National Natural Science Foundation of China under Grants 11171368 and 11171094. The research is also supported by Program for Innovative Research Team (in Science and Technology) in University of Henan Province under Grant IRTSTHN (14IRTSTHN023).

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

## Authors and Affiliations

1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China