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2-Pile Nim with a Restricted Number of Move-Size Imitations

Urban Larsson
  • Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden. E-mail:
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Published Online: 2010-01-26 | DOI: https://doi.org/10.1515/INTEG.2009.053


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, .

Keywords.: Beatty pairs; game with memory; impartial game; move-size dynamic; Muller twist; Nim; Wythoff Nim

About the article

Received: 2008-01-16

Revised: 2009-08-17

Accepted: 2009-08-28

Published Online: 2010-01-26

Published in Print: 2009-12-01

Citation Information: Integers, Volume 9, Issue 6, Pages 671–690, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/INTEG.2009.053.

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Urban Larsson and Simon Rubinstein-Salzedo
International Journal of Game Theory, 2017
Matthew Cook, Urban Larsson, and Turlough Neary
Natural Computing, 2017, Volume 16, Number 3, Page 397
Endre Boros, Vladimir Gurvich, and Vladimir Oudalov
International Journal of Game Theory, 2013, Volume 42, Number 4, Page 891
Urban Larsson
Journal of Combinatorial Theory, Series A, 2013, Volume 120, Number 5, Page 1116

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