Nim-sequence
The Sprague–Grundy theorem implies that a heap of size n is equivalent to a nim heap of a given size, usually noted G(n). The analysis of an octal game then consists in finding the sequence of the nim-values for heaps of increasing size. This sequence G(0), G(1), G(2) ... is usually called the nim-sequence of the game.
All finite octal games analyzed so far have shown a nim-sequence ultimately periodic, and whether all finite octal games are ultimately periodic is an open question. It is listed by Richard Guy as an important problem in the field of combinatorial games.
Read more about this topic: Octal Game