What is the name of the following combinatorial game:

Two players, moving in turn.

Positions: $0,1,2,\ldots$.

Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$ if $n>0$.

No move for $0$ which loses.

The determination of the winning strategy is easy: $n=2^k(2m+1)>0$ wins if and only if $k$ is even.

Determinating the winning strategy for the misere convention ($0$ wins) is also easy:

$2^k$ wins for $k$ odd and loses for $k$ even,

$2^k(2m+1)$ with $m\geq 1$ wins for $k$ even and loses for $k$ odd.

This game has certainly been described somewhere. Does it has a name? Does somebody have a reference?


1 Answer 1


This is "Mark", supposedly due to Mark Krusemeyer; see the first sentence of the introduction to https://arxiv.org/abs/1509.04199 and section 2 of https://doi.org/10.37236/2015.


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