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Consider a game of nim with $3$ heaps, the winning strategy is for a player to leave always an even total number of $1$'s, $2$'s, and $4$'s. (source: http://en.wikipedia.org/wiki/Nim)

How would the winning strategy change if one or both players are allowed to skip a step once at any time?

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If both players have the option, not at all. If I am in a winning position in the original game, I will make the move I would make in the original game. If my opponent passes, I pass. We are now back in the basic game.

If only one player has the option, that player wins. If he is winning, he never passes and just wins. If he is losing, he passes and becomes the winning player.

This is just a start of some of them material in Winning Ways

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  • $\begingroup$ Thank you! I always intend to check out winning ways, but the length intimidates me. I certainly will sooner or later! $\endgroup$ – hattoriace Mar 7 '13 at 20:18
  • $\begingroup$ @hattoriace You might want to try "Lessons in Play" as a more accessible (and shorter) book for the basics. $\endgroup$ – Mark S. Dec 29 '13 at 7:01
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If one player is allowed to skip, he can turn a losing position to a winning position, hence he always wins.

If both are allowed to skip once, the first to skip would play with an opponent who can skip and thus win. Hence it would be a bad idea to skip and the classical Nim strategies apply.

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    $\begingroup$ Great minds think backwards from each other. $\endgroup$ – Ross Millikan Mar 7 '13 at 19:20
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The interesting question, first proposed (I believe) by David Gale is:

"What is the winning strategy if there is a single pass move that can be used by either player?"

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    $\begingroup$ Welcome to MSE! I realize you don't yet have enough reputation, but this is better as a comment. Regards $\endgroup$ – Amzoti Apr 29 '13 at 15:06
  • $\begingroup$ I would suggest you make this into a new question. It is a valid one. $\endgroup$ – Ross Millikan Apr 29 '13 at 15:25
  • $\begingroup$ I was actually thinking about this very question but ended up describing this easier version. $\endgroup$ – hattoriace May 19 '13 at 0:20

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