## Problem of the Week 1133## RoundUpAlice 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:
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 © Copyright 2010 Stan Wagon. Reproduced with permission. |

5 October 2010