# 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.