## MATHS BITE: Wythoff’s Game

Wythoff’s game is a two-player mathematical strategy game. It is played with two piles of counters and goes as follows:

Players take turns to remove counters from one, or both piles (if they are removing stones from both piles, they must remove the same number of stones from each). The game ends when one person removes the last counter, or counters, thus winning.

It is name Wythoff’s game, as it was Dutch mathematician W. A. Wythoff that published a mathematical analysis of the game in 1907, although it was previously know in China as tsyan-shidzi (“choosing stones”).

### Optimal Strategy?

Any position of the game can be described by a pair of integers (x,y) where x≤y. This describes the size of both piles in the position.

The strategy of the game involves cold and hot positions:

• A cold position is one where the player whose turn it is to move will lose, even with his best move;
• A hot position is one where the player whose turn it is to move will win with his best move.

Thus, the optimal strategy for a person in a hot position is to move to a cold position.

We can classify the positions as hot or cold recursively, using 3 simple rules:

1. (0,0) is a cold position;
2. Any position from which a cold position can be reached in a single move is a hot position;
3. If every move leads to a hot position then the position is cold.

For example (taken from Wikipedia):

“all positions of the form (0, y) and (y, y) with y > 0 are hot, by rule 2. However, the position (1,2) is cold, because the only positions that can be reached from it, (0,1), (0,2), and (1,1), are all hot. The cold positions (x, y) with the smallest values of x and y are (0, 0), (1, 2), (3, 5), (4, 7),(6,10) and (8, 13).”

### Connection with the Golden Ratio

Wythoff discovered that the cold positions can be determined using the Golden Ratio. Namely, if n is any natural number, then

is a cold position (where we use the floor function and φ is the Golden Ratio).