2-pile Nim with a Restricted Number of Move-size Imitations

Details 2-pile Nim with a Restricted Number of Move-size Imitations 2-pile Nim with a Restricted Number of Move-size Imitations

2007-10-19

arxiv.org/abs/0710.3632v3

Mathematics

Combinatorics

18 pages, with an appendix by Peter Hegarty

Scientific paper

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a variant with a blocking manoeuvre on $p-1$ diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is.

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