By Ferguson T.S.
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R. K. Guy (1996) Impartial games, in Games of No Chance, MSRI Publications 29, R. , Cambridge University Press, 61-78. Richard K. Guy and Cedric A. B. Smith (1956) The G-values of various games, Proc. Cambridge Philos. Soc. 52, 514-526. Emanuel Lasker (1931) Brettspiele der V¨olker (1931) Berlin, 183-196. H. W. Lenstra (1977-78) Nim multiplication, in Seminaire de Theorie des Nombres, expos´e No. 11, Univ. Botdeaux. E. H. Moore (1910) A generalization of a game called nim, Ann. Math. 11, 93-94.
The colon principle states that the two graphs G1 and G2 have the same Sprague-Grundy value. 4. The claim that G1 and G2 have the same Sprague-Grundy value is equivalent to the claim that the sum of the two games has Sprague-Grundy value 0. In other words, we are to show that the sum G1 + G2 is a P-position. 4 Here is a strategy that guarantees you a win if you are the second player to move in G1 + G2 . If the first player moves by chopping one of the edges in G in one of the games, then you chop the same edge in G in the other game.
One remarkable implication of this theorem is that every progressively bounded impartial game, when considered as a component in a sum of such games, behaves as if it were a nim pile. That is, it may be replaced by a nim pile of appropriate size (its SpragueGrundy value) without changing the outcome, no matter what the other components of the sum may be. We express this observation by saying that every (progressively bounded) impartial game is equivalent to some nim pile. 3 Applications. 1. Sums of Subtraction Games.