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Problem of the Week 1133


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.

© Copyright 2010 Stan Wagon. Reproduced with permission.

5 October 2010