Wythoff's Game

A game played with two heaps of counters in which a player may take any number from either heap or the same number from both. The player taking the last counter wins. The $r$th Safe combination is $(x, x+r)$, where $x=\left\lfloor{\phi r}\right\rfloor$, with $\phi$ the Golden Ratio and $\left\lfloor{x}\right\rfloor$ the Floor Function. It is also true that $x+r=\left\lfloor{\phi^2 r}\right\rfloor$. The first few Safe combinations are (1, 2), (3, 5), (4, 7), (6, 10), ....

See also Nim, Safe

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 39-40, 1987.

Coxeter, H. S. M. ``The Golden Section, Phyllotaxis, and Wythoff's Game.'' Scripta Math. 19, 135-143, 1953.

O'Beirne, T. H. Puzzles and Paradoxes. Oxford, England: Oxford University Press, pp. 109 and 134-138, 1965.