## Problem of the Week 1132## Subtract or HalveAlice and Bob play a game. A positive integer starts the game and the players take turns changing the current value to a smaller one, and then passing the smaller one back to their opponent. On each move, a player may subtract 1 from the integer, or halve it, rounding down 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:
If the game starts with 1132, which player has a winning strategy? Extra credit - Find a complete characterization of the numbers on which Alice wins.
- What is the limiting probability that Alice wins on a random input (where we assume the input is random and less than n, and then let n approach infinity)?
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. |

28 September 2010