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This process gets much more fascinating when the victory condition is not known at the start of the game. The game of Whim is identical to Nim, except that one additional class of move is allowed. At any point, once per game, either player can sacrifice his move to set the win condition, which is then permanent. This has the fun side effect of both choosing the win condition and changing the turn order, forcing an additional move by the opposition. I'd be curious to hear your thoughts on Whim. It looks to me like almost identical analysis can get you most of the way to winning Whim, however, the ability to switch turn order and set the win condition means you must pick a suitable time to do that. That's hard to do, because you opponent might beat you to the punch.