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A professor of one of my courses introduced us to a game to play during down-time, we start with 1000, then we take turns subtracting the powers of 2 (1 to 512), from 1000, we can use the same power of 2 multiple times, but you cannot end up with a negative difference, or a difference of zero. That is, the winner reaches a difference of 1, from which the next player has no moves (cannot subtract 1 from 1 to leave a difference of zero).
He asked us if we could have any strategies and if it matters what player goes first. Now, I have simulated many games and come to conclusion that it might not matter who goes first, but any strategy I seem to come up with seems flawed.
Has anyone else heard of this game? Is there a winning strategy, and does it matter who goes first or not? Furthermore, is there any generalized version of this game, say maybe with powers of n, or starting with an integer total m?