Problem of the Week 1133

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?

Source: Mark Krusemeyer, Carleton College, who observes that it will likely appear in a forthcoming problem book, a successor to the Wohascum County Problem Book. It also appeared recently in the Buhler-Berlekamp problem column for MSRI's Emissary newletter.

5 October 2010