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  • Bulletin of the Australian Mathematical Society, Volume 86, Issue 3
  • December 2012, pp. 506-509


  • GRANT CAIRNS (a1) and NHAN BAO HO (a2)
  • DOI:
  • Published online: 12 June 2012

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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[2]G. Cairns and N. B. Ho , ‘Min, a combinatorial game having a connection with prime numbers’, Integers 10 (2010), 765770.

[4]G. Cairns , N. B. Ho and T. Lengyel , ‘The Sprague–Grundy function of the real game Euclid’, Discrete Math. 311 (2011), 457462.

[5]A. J. Cole and A. J. T. Davie , ‘A game based on the Euclidean algorithm and a winning strategy for it’, Math. Gaz. 53 (1969), 354357.

[7]G. Nivasch , ‘The Sprague–Grundy function of the game Euclid’, Discrete Math. 306(21) (2006), 27982800.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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