Guaranteed successful strategies for a square achievement game on an n by n grid

TL;DR

This paper analyzes a square achievement game on an n by n grid, providing guaranteed winning or non-losing strategies for players for specific grid sizes, supported by a computer program.

Contribution

It offers proven strategies ensuring the second player's non-loss for n=3,4 and the first player's win for n=5, advancing understanding of optimal play in this game.

Findings

Second player can avoid losing at n=3 and n=4

First player can guarantee a win at n=5

Computer program SQRGAME2 supports strategy proofs

Abstract

At some places (see the references) Martin Erickson describes a certain game: "Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner." Then he asks "What is the outcome of the game given optimal play?" or "What is the smallest n such that the first player has a winning strategy?" For n lower than 3 a win is obviously impossible. The aim of this article and the additionally (in the source package) provided computer program SQRGAME2 is to give and prove sure strategies for the second player not to lose if n is 3 or 4, and for the first player to win if n is 5.