International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 40796, 8 pages

One-pile misère Nim for three or more players

Annela R. Kelly

Department of Mathematics, Roger Williams University, Bristol 02809, RI, USA

Received 29 July 2004; Revised 5 May 2006; Accepted 9 May 2006

Copyright © 2006 Annela R. Kelly. 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.


The analysis of the classical game of Nim relies on binary representation of numbers as shown in the books of Berlekamp and Gardener. There is considerable interest in generalizations and modifications of the game. We will consider one-pile misère Nim for more than two players. In the case of three or more players, the impartial game theory results rarely apply. In this note, we analyze the game in a variety of cases where alliances are formed among the players.