Table of Contents
International Journal of Combinatorics
Volume 2011, Article ID 824742, 9 pages
Research Article

Some Remarks on End-Nim

Department of Mathematics, La Trobe University, Melbourne, VIC 3086, Australia

Received 26 September 2011; Accepted 14 November 2011

Academic Editor: Johannes Hattingh

Copyright © 2011 Grant Cairns and Nhan Bao Ho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.