2-Pile Nim With a Restricted number of Move-size Imitations (with an appendix by Peter Hegarty)

Journal article, 2009

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. The paper includes an appendix by Peter Hegarty, Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, hegarty@chalmers.se.