Grundy's game

The game ends when only heaps of size two and smaller remain, none of which can be split unequally.

In each of these cases, player 1 can ensure that on the next move he hands back to his opponent a heap of size 4 plus heaps of size 2 and smaller: Now player 2 has to split the 4-heap into 3 + 1, and player 1 subsequently splits the 3-heap into 2 + 1: The game can be analysed using the Sprague–Grundy theorem.

This mapping is captured in the On-Line Encyclopedia of Integer Sequences as OEIS: A002188: Using this mapping, the strategy for playing the game Nim can also be used for Grundy's game.

Whether the sequence of nim-values of Grundy's game ever becomes periodic is an unsolved problem.

Elwyn Berlekamp, John Horton Conway and Richard Guy have conjectured[1] that the sequence does become periodic eventually, but despite the calculation of the first 235 values by Achim Flammenkamp, the question has not been resolved.

Stacks of coins. Any of these stacks can be split into two stacks of different sizes: once the leftmost stack of three has been split, it can be split no further.