#
Problem 7: Wythoff's Game and Fibonacci Word

Wythoff's game, a variant of the game of Nim, is a
two-player game of strategy.
It is played with two piles of tokens, one being
initially non-empty.
Players take turns removing either a positive number of
tokens from one pile or the same number of tokens from both piles.
When there are no tokens left, the game ends and the last player is the winner.

A configuration of the game is described by a pair of natural numbers
$(m,n)$, $m\leq n$, where $m$ and $n$ are the number of tokens on the two
piles.
Note that $(0,n)$ as well as $(n,n)$, $n\gt 0$, are winning
configurations.
The smallest losing configuration is $(1,2)$ and then all
configurations of the form $(m+1,m+2)$, $(1,m)$ and $(2,m)$ for $m\gt 0$ are
winning configurations.

It is known that losing configurations follow a regular pattern determined by
the golden ratio.
Thus, the following question.

Is there any close relation between Wythoff's game and the infinite Fibonacci
word?

## References

W. A. Wythoff. A modification of the game of Nim. *Nieuw Arch. Wisk.*,
**8**:199-202, 1907/1909.