Introduction to the Prisoners Versus Guards Game

TL;DR

This paper introduces a strategic two-player game involving prisoners and guards on an n-by-n checkerboard, analyzing optimal strategies and maximum prisoner placements with mathematical bounds and conjectures.

Contribution

It defines a new game, analyzes winning strategies for small boards, and provides bounds and conjectures for maximum prisoner placements.

Findings

Maximum prisoners less than (7n^2+4n)/11

Winning strategies identified for small n<5

Conjecture: maximum approximately 3n^2/5+O(n)

Abstract

We introduce a two-player game in which one and his/her opponent attempt to pack as many ``prisoners'' as possible on the squares of an n-by-n checkerboard; each prisoner has to be ``protected'' by at least as many guards as the number of the other prisoners adjacent. Initially, the board is covered entirely with guards. The players take turns adjusting the board configuration using one of the following rules in each turn: I. Replace one guard with a prisoner of the player's color. II. Replace one prisoner of either color with a guard and replace two other guards with prisoners of the player's color. We analyze winning strategies for small n (n<5) and the maximum number of prisoners in general. We show that this maximum is less than (7n^2+4n)/11 and conjecture it is more likely 3n^2/5+O(n).