Winning strategy

Question

Is there a winning strategy for player one or two in the following scenario: The game begins with the number 2012. In one turn, a player can subtract from the current number any natural number less than or equal to it that is a power of 2. The player who reaches 0 wins.

Answer 1

It seems a player loses if the number is a multiple of 3 on his/her turn. Indeed, in that situation it is impossible to subtract a power of 2 and obtain a difference that is again a multiple of 3. On the other hand, if the number is not a multiple of 3, then the player can force the difference to become a multiple of 3 by playing either 1 or 2.

In particular, the first player wins in this game, where the starting number 2012 is not a multiple of 3. He/she starts by playing 2 (or 512 if he/she wants to go a little faster) and then just makes sure to make the difference a multiple of 3 at the end of all his/her turns.