On: The Putnam Mathematical Competition’s Unsolved Problem

[chriscool]

Now a good strategy for Alice is to first play in the middle, and after that use the following rules:

- if Bob removes one stone and the result is that it adds a stone both on the left and right side of the board, then Alice replays the stone that Bob just removed,

- otherwise whatever Bob plays, Alice plays the same thing on the other side of the board (symmetrically regarding the middle of the board)

This ensures that:

5) the situation is always symmetrical after Alice played

6) when Bob plays once and then Alice plays once, if the nb of stone on the board increased since before Bob just played, then it increased by an even amount

[chriscool → chriscool]

As after Alice first played in the middle, there was just 1 stone, then because of 6):

7) when it's Bob turn to play, the number of stones on the board is always odd

Alice wins because of 7) and 4)!