Abstract
In this paper we prove that in a Quasi-Dawson’s Chess (a restricted version of Dawson’s Chess) playing on a 3 × d board, the first player is loser if and only if d (mod)5 = 1 or d (mod)5 = 2. Furthermore, we have designed two algorithms that are responsible for storing the results of Quasi-Dawson’s Chess games having less than d + 1 files and finding the strategy that leads to win, if there is a possibility of winning (by a wining position, we mean one from which one can win with best play). Moreover we show that the total complexity of our algorithms is O(d 2). Finally we have implemented our algorithm in C++ which admits the main results of the paper even for large values of d.
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References
M. H. Albert, R. J. Nowakowski and D. Wolfe, Lessons in play: An introduction to the combinatorial theory of games, Peters, Wellesley (2007).
E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning ways for your mathematical plays. Vol. 1: Games in general, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), London-New York (1982).
J. H. Conway, On numbers and games, Second edition, Peters, Ltd., Natick, MA (2001).
T. R. Dawson, Caissas wild roses, 1935, in: Five Classics of Fairy Chess, Dover Publications, Inc. (1973).
T. S. Ferguson, A note on Dawson’s chess, Unpublished, http://www.miseregames.org/docs/DawsonChess.pdf.
P. M. Grundy, Mathematics and games, Eureka 2 (1939), 6–8; reprinted 27 (1964), 9–11.
P. M. Grundy and C. A. B. Smith, Disjunctive games with the last player losing, Proc. Cambridge Philos. Soc, 52 (1956), 527–533.
R. K. Guy, C. A. B. Smith, The G-values of various games, Proc. Cambridge Philos. Soc, 52 (1956), 514–526.
A. Guo, E. Miller, and M. Weimerskirch, Potential applications of commutative algebra to combinatorial game theory, in Kommutative Algebra, abstracts from the April 19–25, 2009 workshop, organized by W. Bruns, H. Flenner, and C. Huneke, Oberwolfach rep. 22 (2009), 23–26.
E. Miller, Theory and applications of lattice point methods for binomial ideals, in Combinatorial aspects of commutative algebra and algebraic geometry, Proceedings of Abel Symposium held at Voss, Norway, 1–4 June 2009, Abel Symposia, vol. 6 (2011), 99–154, Springer, Berlin-Heidelberg.
T. E. Plambeck, Advances in losing, In Games of no chance 3, papers from the Combinatorial Game Theory Workshop held in Banff, AB, June 2005, edited by Michael H. Albert and Richard J. Nowakowski, MSRI Publications, Cambridge University Press, Cambridge, forthcoming. arXiv:math.CO/0603027
T. E. Plambeck, A. N. Siegel, Misère quotients for impartial games, J. Combin. Theory Ser. A, 115(4) (2008), 593–622. arXiv:math.CO/0609825v5
A. N. Siegel, Misère games and misère quotients, unpublished lecture notes. arXiv: math.CO/0612616
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The first author is thankful to the National Elite Foundation of Iran for financial support.
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Borna, K., Ashrafi Payaman, N.A. Dawson’s chess revisited. Indian J Pure Appl Math 44, 771–794 (2013). https://doi.org/10.1007/s13226-013-0042-7
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DOI: https://doi.org/10.1007/s13226-013-0042-7