2-Pile Nim With a Restricted number of Move-size Imitations (with an appendix by Peter Hegarty)
Urban Larsson · Chalmers Publication Library (Chalmers University of Technology) · 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 justremoved say x > 0 tokens from the shorter pile (either pile in case they havethe 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 therules of the game, neither player is allowed to imitate his opponent on more thanp−1 consecutive moves. We prove that the strategy of this game resembles closelythat of a variant of Wythoff Nim—a variant with a blocking manoeuvre on p − 1diagonal positions. In fact, we show a slightly more general result in which wehave relaxed the notion of what an imitation is. The paper includes an appendixby Peter Hegarty, Mathematical Sciences, Chalmers University of Technology andUniversity of Gothenburg, [email protected].