# A Rook Game

Problem: Two players take turns moving a rook on an 8×8 chessboard. The rook is only allowed to move south or west (but not both in a single turn), and may move any number of squares in the chosen direction on a turn. The loser is the player who first cannot move the rook. What is the optimal play for any starting position?

Solution: Take advantage of the symmetry of the board. If the rook is not on the diagonal, the optimal strategy is to move it to the diagonal. Then when the other player moves it off, your next move is to move it back to the diagonal. If your opponent starts their turn with the rook always on the diagonal, then you will never lose, and by the symmetry of the board you can always move the rook back to the diagonal. This provides an optimal algorithm for either player. In particular, if the rook starts on a square that is not on the diagonal, then player 1 can guarantee a win, and otherwise player 2 can.

Symmetry is one of the most powerful tools in all of mathematics, and this is a simple albeit illustrative example of its usage.

## 2 thoughts on “A Rook Game”

1. This can be seen as a game of nim with 2 piles, heh.

Like

2. Dmitriy

Not really insightful. Just imagine the same game but on on 3d board (that is, the board with 8x8x8 cubes) and this method no longer works. Looks more like a trick for this special case.

Nayuki is right: this is an easy example where combinatorial game theory could be applied.

Like