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We know how to win the classic regular Nim (two players) Classic rules: Any number of beans into any number of separate piles Each move, the player whose turn it is, must choose one pile of beans and remove anywhere from one bean to all the beans in only ONE pile Player taking the last bean looses (as example).

I am looking for the correct strategy for the following variation: Same rules as in the classic regular Nim EXCEPT THAT only once in the game, a player, and only one, MAY PASS his turn.

I found some winning positions 2 1,1 1,N,N (with N>1) 2,3,5 4,5,8, 6,7,3

and I believe that, before the Pass is used, we need to use one "virtual" pile with one bean but I could not find a general strategy.

  • 2
    What's the difference between the modified game and the original game with one additional one-bean pile?2011-04-17
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    Do you mean that each player may pass once, or once one player has passed, no other player can pass?2011-04-17
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    I suppose if each player can pass once, then whoever has a winning strategy in the classic game has a winning strategy in the new game, namely: Pass if your opponent passed last time, otherwise play your winning move. On the other hand, @Alon's comment shows how you can translate a "one person can pass once" game into a classic game.2011-04-17
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    This is still an impartial game. If you follow the proof in http://www.amazon.com/Winning-Ways-Your-Mathematical-Plays/dp/1568811306 that every impartial game is a nim-heap, the pass is clearly a one bean heap.2011-04-18

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