Arithmetic series

A running total starts at 0. Two players alternate adding 1-10 to the running total. The player who makes the running total 100 is the winner.

The first player can always win. The strategy is to start by adding 1. Then, subtract 11 from your opponents moves, repeatedly, until you reach 100. This works because 100 = 1 + 99, and 99 is a multiple of 11. If the first player messes up, and the second player is able to reach 1 plus some multiple of 11, then the second player can win the game.

If the running total started at some number other than 0, for which numbers would the second player be guaranteed to win? Answer: Any number which is not a multiple of 11.

How can we generalize this game? Suppose that the running total starts at t, and players alternate adding 1-u to the running total. The player who makes the running total v is the winner. What are the conditions on which the first player is guaranteed to win? The second player?


Here's where this idea came from.