RoundUp

Alice and Bob play a game. A positive integer starts the game and the players take turns changing the current value and passing the new number back to their opponent.

On each move, a player may subtract 1 from the integer, or halve it, rounding up if necessary. The person who first reaches 0 is the winner.

Alice goes first: she makes her choice of move on the starting value.

For example, starting at 15 a legal game (if not particularly well played) could be:


Alice .......................................15 → 8
Bob ......................................... 8 → 7
Alice ....................................... 7 → 4
Bob ......................................... 4 → 2
Alice ....................................... 2 → 1
Bob ......................................... 1 → 0 Bob wins.

For which values of n is there a winning strategy for Alice?

http://mathforum.org/wagon/fall10/p1133.html