The two-move chess game has the same rules as the regular one, with only one exception. Each player has to make two consecutive moves at a time. Does the White player, who makes the first two moves, have a non-losing strategy?

Sunday, August 5, 2012

Please email your solutions to puzzles@cotpi.com.

### 2 comments

### Credit

This puzzle is taken from:

- Savchev, Svetoslav; Andreescu, Titu. Mathematical Miniatures. Washington, D.C.: Mathematical Association of America, 2003. 1.

## Tim Ophelders solved this puzzle:

In two-move chess, the White player can play the moves Na3 followed by Nb1, resulting in the starting position, but this time with the opponent to move. Assume the White player does not have a non-losing strategy, then neither does the opponent. So both players would lose the game; but since there can only be one losing player in a game of chess, this is a contradiction. Thus, the White player has a non-losing strategy.

## Ryan Batterman solved this puzzle:

I will refer to a sequence of two consecutive moves made by a single player as a 2-move.

Suppose, for the purpose of contradiction, that the White player does not have a non-losing strategy. Hence, for every initial 2-move that the White player makes, the Black player has a strategy which eventually defeats the White player.

Consider the following initial 2-move made by the White player: g1-f3 and f3-g1. Note that this 2-move, which just moves a knight forward and back, does not alter the board. In other words, after this initial 2-move, the Black player has the exact same position as the White player had before making his move.

Since in the initial position the player that did not make the first 2-move had a forced win, and since this new position is essentially the same as the original one except that the board is flipped, we can conclude that in this new position, the player not making the 2-move, i.e. the White player, has a forced win.

Therefore, we assumed that the Black player has a forced win and concluded that the White player has a forced win as well. This is a contradiction. Hence, our assumption must be wrong. Thus, the White player does have a non-losing strategy.