For each move of g 2 g 3 there is an associated move

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For each move of G 2 + G 3 there is an associated move in G 1 + G 3 : If one of the players moves in G 3 when playing G 2 + G 3 , this corresponds to the same move in G 3 when playing G 1 + G 3 . If one of the players plays in G 2 when playing G 2 + G 3 , say by moving from a Nim pile with y chips to a Nim pile with z < y chips, then the corresponding move in G 1 + G 3 would be to move in G 1 from a position with Sprague-Grundy value y to a position with Sprague-Grundy value z (such a move exists by the definition of the Sprague-Grundy function). There may be extra moves in G 1 + G 3 that do not correspond to any move G 2 + G 3 , namely, it may be possible to play in G 1 from a position with Sprague-Grundy value y to a position with Sprague-Grundy value z>y . When playing in G 1 + G 3 , player A can pretend that the game is really G 2 + G 3 . If player A’s winning strategy is some move in G 2 + G 3 , then A can play the corresponding move in G 1 + G 3 , and pretends that this move was made in G 2 + G 3 . If A’s opponent makes a move in G 1 + G 3 that corresponds to a move in G 2 + G 3 , then A pretends that this move was made in G 2 + G 3 . But player A’s opponent could also make a move in G 1 + G 3 that does not correspond to any move of G 2 + G 3 , by moving in G 1 and increasing the Sprague-Grundy value of the position in G 1 from y to z>y . In this case, by the definition of the Sprague-Grundy value, player A can simply play in G 1 and move to a position with Sprague-Grundy value y . These two turns correspond to no move, or a pause, in the game G 2 + G 3 . Because G 1 + G 3 is progressively bounded, G 2 + G 3 will not remain paused forever. Since player A has a winning strategy for the game G 2 + G 3 , player A will win this game that A is pretending to play, and this will correspond to a win in the game
1.1 Impartial games 27 G 1 + G 3 . Thus whichever player has a winning strategy in G 2 + G 3 also has a winning strategy in G 1 + G 3 , so G 1 and G 2 are equivalent games. We can use this theorem to find the P - and N -positions of a particular impartial, progressively bounded game under normal play, provided we can evaluate its Sprague-Grundy function. For example, recall the 3-subtraction game we considered in Example 1.1.10. We determined that the Sprague-Grundy function of the game is g ( x ) = x mod 4. Hence, by the Sprague-Grundy theorem, 3-subtraction game with starting position x is equivalent to a single Nim pile with x mod 4 chips. Recall that (0) P Nim while (1) , (2) , (3) N Nim . Hence, the P -positions for the Subtraction game are the natural numbers that are divisible by four. Corollary 1.1.1. Let G 1 and G 2 be two progressively bounded impartial combinatorial games under normal play. These games are equivalent if and only if the Sprague-Grundy values of their starting positions are the same.

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