Three-player Domineering

Authors: Alessandro Cincotti

Abstract:

Domineering is a classic two-player combinatorial game usually played on a rectangular board. Three-player Domineering is the three-player version of Domineering played on a three dimensional board. Experimental results are presented for x×y ×z boards with x + y + z < 10 and x, y, z ≥ 2. Also, some theoretical results are shown for 2 × 2 × n board with n even and n ≥ 4.

Keywords: Combinatorial games, Domineering, three-playergames.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333823

References:

[1] E.R. Berlekamp, "Blockbusting and Domineering," Journal of Combinatorial Theory Ser. A, vol. 49, pp. 67-116, 1988.
[2] E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning way for your mathematical plays, 2nd ed. Natick, Massachusetts: A K Peters, 2001.
[3] D.M. Breuker, J.W.H.M. Uiterwijk, H.J. van den Herik, "Solving 8 × 8 Domineering," Theoretical Computer Science, vol. 230, pp. 195-206, 2000.
[4] N. Bullock, "Domineering: Solving Large Combinatorial Search Spaces," ICGA Journal, vol. 25, no. 2, pp. 67-84, 2002.
[5] A. Cincotti, "Three-player partizan games," Theoretical Computer Science, vol. 332, pp. 367-389, 2005.
[6] A. Cincotti, "The Game of Maundy Block," Proceedings of World Academy of Science, Engineering and Technology, vol. 26, pp. 74-77, 2007.
[7] A. Cincotti, "Three-player Hackenbush played on strings is NPcomplete," Proceedings of the IAENG International Conference on Computer Science (ICCS-08), pp. 226-230, 2008.
[8] A. Cincotti, "The Game of Cutblock," INTEGERS: Electronic Journal of Combinatorial Number Theory, vol. 8, #G06, 2008.
[9] J. H. Conway, On Numbers and Games, 2nd ed. Natick, Massachusetts: A K Peters, 2001.
[10] M. Gardner, "Mathematical games," Scientific American, vol. 230, pp. 106-108, 1974.
[11] M. Lachmann, C. Moore, I. Rapaport, "Who Wins Domineering on Rectangular Boards," in More Games of No Chance, R.J. Nowakowski, Ed. Cambridge University Press, 2002, pp. 307-315.
[12] J. Propp, "Three-player impartial games," Theoretical Computer Science, vol. 233, pp. 263-278, 2000.