Some Remarks on End-Nim

We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the wi...

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Bibliographic Details
Published in:International journal of combinatorics 2011-12, Vol.2011 (2011), p.1-9
Main Authors: Cairns, Grant, Ho, Nhan Bao
Format: Article
Language:English
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Summary:We reexamine Albert and Nowakowski's variation on the game of Nim, called End-Nim, in which the players may only remove coins from the leftmost or rightmost piles. We reformulate Albert and Nowakowski's solution to this game. We examine its misère version and a further variant where the winner is the player who reduces the game to a single pile; we call this Loop-End-Nim. We show that the three games, End-Nim, misère-End-Nim, and Loop-End-Nim, all have the same losing positions, except for the positions where all the piles are of equal size. We also give some partial results concerning the higher Sprague-Grundy values of the three games.
ISSN:1687-9163
1687-9171
DOI:10.1155/2011/824742