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 th safe combination is , where , with  the golden ratio and  the floor function. It is also true that . The first few safe combinations are (1, 2), (3, 5), (4, 7), (6, 10), ... (OEIS A000201 and A001950), which are the pairs of elements from the complementary Beatty sequences for  and  (Wells 1986, p. 40).

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.

Sloane, N. J. A. Sequences A000201/M2322 and A001950/M1332 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986.

Wythoff, W. A. "A Modification of the Game of Nim." Nieuw Arch. Wisk. 8, 199-202, 1907/1909.