MathDB

In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible

n\times n

matrices with entries in the field

\mathbb{Z}/p\mathbb{Z}

of integers modulo

p,

where

n

is a fixed positive integer and

p

is a fixed prime number. The rules of the game are:
(1) A player cannot choose an element that has been chosen by either player on any previous turn.
(2) A player can only choose an element that commutes with all previously chosen elements.
(3) A player who cannot choose an element on his/her turn loses the game.
Patniss takes the first turn. Which player has a winning strategy?

Problem Statement