Wythoff's game is a two-player game with two heaps whose sizes are nonnegative integers. A player must take a positive number of counters from either heap or
the same positive number from both, and the player taking the last counter wins.
With the smaller heap first, the safe combinations are for integers, where , with the golden ratio and the floor
function. It is also true that . The first few nonzero 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).
The remoteness
counts the number of moves when the winning player wins as quickly as possible and
the losing player delays defeat as long as possible. For , Sánchez Galán and Xu (2026) proved
where
and, for ,
is the unique positive
integer
with
or . Also, when , and otherwise. This extends the losing-position remoteness
values of Boros et al. (2024) to every position. Sánchez Galán
and Xu (2026) report ChatGPT assistance with writing the supplementary verification
code.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 39-40, 1987.Bogomolny,
A. "Wythoff's Nim." https://cut-the-knot.org/pythagoras/withoff.shtml.Boros,
E.; Gurvich, V.; Makino, K.; and Vyalyi, M. "Computing Remoteness Functions
of Moore, Wythoff, and Euclid's Games." Internat. J. Game Theory53,
1315-1333, 2024. https://doi.org/10.1007/s00182-024-00914-2.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.Sánchez Galán, R. and Xu, Y. "A Remoteness
Formula for Generalized Wythoff Games." 21 Oct 2026. https://arxiv.org/abs/2610.30427.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.
for integers 
, where 
, with 
the golden ratio and 
the floor
function. It is also true that 
. The first few nonzero 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).
counts the number of moves when the winning player wins as quickly as possible and
the losing player delays defeat as long as possible. For 
, Sánchez Galán and Xu (2026) proved
and, for 
,
is the unique positive
integer 
with 
or 
. Also, 
when 
, and 
otherwise. This extends the losing-position remoteness
values of Boros et al. (2024) to every position. Sánchez Galán
and Xu (2026) report ChatGPT assistance with writing the supplementary verification
code.
