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A RESTRICTION OF EUCLID

  • GRANT CAIRNS (a1) and NHAN BAO HO (a2)

Abstract

Euclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.

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Copyright

Corresponding author

For correspondence; e-mail: G.Cairns@latrobe.edu.au

References

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[1]Cairns, G. and Ho, N. B., ‘Ultimately bipartite subtraction games’, Australas. J. Combin. 48 (2010), 213220.
[2]Cairns, G. and Ho, N. B., ‘Min, a combinatorial game having a connection with prime numbers’, Integers 10 (2010), 765770.
[3]Cairns, G. and Ho, N. B., ‘Some remarks on End-Nim’, Int. J. Comb. 2011 (2011), Art. ID 824742, 9 pp.
[4]Cairns, G., Ho, N. B. and Lengyel, T., ‘The Sprague–Grundy function of the real game Euclid’, Discrete Math. 311 (2011), 457462.
[5]Cole, A. J. and Davie, A. J. T., ‘A game based on the Euclidean algorithm and a winning strategy for it’, Math. Gaz. 53 (1969), 354357.
[6]Grossman, J. W., ‘A nim-type game, problem #1537’, Math. Mag. 70 (1997), 382.
[7]Nivasch, G., ‘The Sprague–Grundy function of the game Euclid’, Discrete Math. 306(21) (2006), 27982800.
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