Advertisement

International Journal of Game Theory

, Volume 47, Issue 2, pp 673–693 | Cite as

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).

References

  1. Albert MH, Nowakowski RJ (2001) The game of End-Nim. Electr J Combin 8:Article R1Google Scholar
  2. Albert MH, Nowakowski RJ (2004) Nim restrictions. Int Electron J Combin Number Theory 4:Article G01Google Scholar
  3. Bouton CL (1901) Nim, a game with a complete mathematical theory. Ann Math 3:35–39CrossRefGoogle Scholar
  4. Cincotti A (2010) \(N\)-player partizan games. Theor Comput Sci 411:3224–3234CrossRefGoogle Scholar
  5. Flammenkamp A, Holshouser A, Reiter H (2003) Dynamic one-pile blocking Nim. Electr J Combin 10:Article N4Google Scholar
  6. Fraenkel AS, Lorberbom M (1991) Nimhoff games. J Combin Theory Ser A 58:1–25CrossRefGoogle Scholar
  7. Friedman E (2000) Variants of Nim. Math Magic Game Arch. http://www2.stetson.edu/~efriedma/mathmagic/archivegame.html. Accessed Nov 2000
  8. Guy RK, Nowakowski RJ (2009) Unsolved problems in combinatorial games. In: Games of no chance 3, Math. Sci. Res. Inst. Publ, vol 56. Cambridge Univ Press, Cambridge, pp 475–500Google Scholar
  9. Holshouser A, Reiter H, Rudzinski J (2003) Dynamic one-pile Nim. Fibonacci Q 41(3):253–262Google Scholar
  10. Kelly AR (2006) One-pile misère Nim for three or more players. Int J Math Math Sci 2006:1–8.  https://doi.org/10.1155/IJMMS/2006/40796 CrossRefGoogle Scholar
  11. Kelly AR (2011) Analysis of one pile misère Nim for two alliances. Rock Mt J Math 41(6):1895–1906CrossRefGoogle Scholar
  12. Krawec WO (2012) Analyzing \(n\)-player impartial games. Int J Game Theory 41:345–367CrossRefGoogle Scholar
  13. Krawec WO (2015) \(n\)-Player impartial combinatorial games with random players. Theor Comput Sci 569:1–12CrossRefGoogle Scholar
  14. Li SYR (1978) \(N\)-person Nim and \(n\)-person Moore’s Games. Int J Game Theory 7:31–36CrossRefGoogle Scholar
  15. Liu WA, Zhao X (2016) Nim with one or two dynamic restrictions. Discr Appl Math 198:48–64CrossRefGoogle Scholar
  16. Loeb DE (1996) Stable winning coalitions. In: Nowakowski RJ (ed) Proc MSRI workshop on combinatorial games, games of no chance 29:451–471. Cambridge University Press, CambridgeGoogle Scholar
  17. Morrison RE, Friedman EJ, Landsberg AS (2012) Combinatorial games with a pass: a dynamical systems approach. arXiv:1204.3222
  18. Low RM, Chan WH (2015) An atlas of N- and P-positions in ‘Nim with a pass’. Int Electron J Combin Number Theory 15:Article G02Google Scholar
  19. Propp J (1996) Three-player impartial games. Theor Comput Sci 233:263–278CrossRefGoogle Scholar
  20. Straffin PD (1985) Three-person winner-take-all games with Mc-Carthys revenge rule. Coll J Math 16:386–394CrossRefGoogle Scholar
  21. Zhao X, Liu WA (2016) One pile misère bounded Nim with two alliances. Discr Appl Math 214:16–33CrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations