A351630 Nim values that occur at infinitely many heap sizes in the combinatorial game Mem0.

0, 12, 1270, 105161

Offset: 0

Author

Aaron N. Siegel, Jun 01 2022

Keywords

• nonn
• hard
• more

Comments

The impartial combinatorial game Mem0 (aka Short Local Nim) is played with heaps of tokens, as in Nim. On each turn, k tokens may be removed from a heap H, provided that k is not equal to the number of tokens that were removed on the immediately preceding move on H.

A heap may be denoted by n_k, where n is the number of tokens remaining and k the number removed on the preceding move. There are many nim values m that occur at just finitely many heap sizes, in the sense that G(n_k) = m for just finitely many choices of n. This sequence gives the exceptional values of m that occur at infinitely many heap sizes.

It is unknown whether there are infinitely many such m. It is remarkable that such simple, parameterless rules give rise to an unusual and mysterious integer sequence.

References

• R. K. Guy and R. J. Nowakowski, Unsolved Problems in Combinatorial Games, More Games of No Chance, MSRI Publications, Volume 42, 2002, pp. 457-473, problem 22.

Links

• Urban Larsson, Simon Rubinstein-Salzedo, and Aaron N. Siegel, Memgames, arXiv:1912.10517 [math.CO], 2019.

Crossrefs

Cf. A131469.