A cellular automaton for blocking queen games

Abstract

We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as Blocking Wythoff Nim, consists of moving a queen as in chess, but always towards (0, 0), and it may not be moved to any of \(k-1\) temporarily “blocked” positions specified on the previous turn by the other player. The game ends when a player wins by blocking all possible moves of the other player. The value of k is a parameter that defines the game, and the pattern of winning positions can be very sensitive to k. As k becomes large, parts of the pattern of winning positions converge to recurring chaotic patterns that are independent of k. The patterns for large k display an unprecedented amount of self-organization at many scales, and here we attempt to describe the self-organized structure that appears. This paper extends a previous study (Cook et al. in Cellular automata and discrete complex systems, AUTOMATA 2015, Lecture Notes in Computer Science, vol 9099, pp 71–84, 2015), containing further analysis and new insights into the long term behaviour and structures generated by our blocking queen cellular automaton.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Notes

  1. 1.

    The periods \((5,1)\) and \((-1,2)\) are for the épaulette below and to the left of the main diagonal which is of course symmetric with the épaulette above and to the right of the main diagonal.

  2. 2.

    We thank Michal Szabados for pointing this out to us.

References

  1. Cook M (2004) Universality in elementary cellular automata. Complex Syst 15(1):1–40

    MathSciNet  MATH  Google Scholar 

  2. Cook M, Larsson U, Neary T (2015) A cellular automaton for blocking queen games. In: Kari J (ed) Cellular automata and discrete complex systems, AUTOMATA 2015, Lecture Notes in Computer Science, vol 9099, pp 71–84

  3. Cook M, Larsson U, Neary T (NA) Generalized cyclic tag systems, with an application to the blocking queen game. In preparation

  4. Fink A (2012) Lattice games without rational strategies. J Comb Theory A 119(2):450–459

    MathSciNet  Article  MATH  Google Scholar 

  5. Gavel H, Strimling P (2004) Nim with a modular Muller twist. Integers 4(G4)

  6. Gurvich V (2010) Further generalizations of Wythoff’s game and minimum excludant function. Tech. Rep. RUTCOR Research Report, 16-2010, Rutgers University

  7. Hegarty P, Larsson U (2006) Permutations of the natural numbers with prescribed difference multisets. Integers 6(A3)

  8. Holshouser A, Reiter H (2001) Problems and solutions: problem 714 (Blocking Nim). Coll Math J 32(5):382

    Google Scholar 

  9. Holshouser A, Reiter H (NA) Three pile Nim with move blocking. http://citeseer.ist.psu.edu/470020.html

  10. Kari J, Szabados M (2015) An algebraic geometric approach to multidimensional words. In: Maletti A (ed) Algebraic Informatics, CAI 2015, Lecture Notes in Computer Science, vol 9270, pp 29–42

  11. Larsson U (2009) 2-pile Nim with a restricted number of move-size imitations. Integers 9(6):671–690

    MathSciNet  Article  MATH  Google Scholar 

  12. Larsson U (2011) Blocking Wythoff Nim. Electron J Comb 18, paper p120

  13. Larsson U (2012) A generalized diagonal Wythoff Nim. Integers 12(G2)

  14. Larsson U (2013) Impartial games emulating one-dimensional cellular automata and undecidability. J Comb Theory A 120:1116–1130

    MathSciNet  Article  MATH  Google Scholar 

  15. Larsson U (2015) Restrictions of m-Wythoff Nim and p-complementary beatty sequences. In: Nowakowski RJ (ed) More games of no chance 4, Proceedings BIRS Workshop on combinatorial Games, 2008, Cambridge University Press, MSRI Publications, vol 63, pp 137–160

  16. Larsson U, Wästlund J (2013) From heaps of matches to the limits of computability. Electron J Comb 20, paper p41

  17. Post EL (1943) Formal reductions of the general combinatorial decision problem. Am J Math 65(2):197–215

    MathSciNet  Article  MATH  Google Scholar 

  18. Smith F, Stănică P (2002) Comply/constrain games or games with a Muller twist. Integers 2(G4)

  19. Wythoff WA (1907) A modification of the game of Nim. Nieuw Arch Voor WisKd 7:199–202

    MATH  Google Scholar 

Download references

Acknowledgments

Urban Larsson is supported by the Killam Trust. Turlough Neary is supported by Swiss National Science Foundation Grants 200021-141029 and 200021-153295.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Turlough Neary.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cook, M., Larsson, U. & Neary, T. A cellular automaton for blocking queen games. Nat Comput 16, 397–410 (2017). https://doi.org/10.1007/s11047-016-9581-2

Download citation

Keywords

  • Wythoff Nim
  • Blocking Wythoff Nim
  • Cellular automata
  • Self-organization