Another Form of Matrix Nim

Thomas S. FERGUSON

University of California at Los Angeles

Abstract:

A new form of matrix nim is proposed and investigated. The positions are m by n matrices of nonnegative integers, where m and n are fixed positive integers. A move consists in choosing a row or column and subtracting some positive integer, k, from each integer of the chosen row or column. The terminal positions are the matrices with at least one zero in every row and column. Last to move wins. The case $m=1$ and $n=2$ is Wythoff's nim. This is the impartial version of the game. There is also the partizan version in which Left is restricted to choosing a coLumn and Right is restricted to choosing a Row. The outcomes of all 2 by 2 positions are found in both the impartial and partizan cases. Some hope is given of being able to solve sums of 2 by 2 games in the partizan case.