When Rooks Miss: Probability through Chess

TL;DR

This paper investigates the probability that a random placement of $n$ rooks on an $n\times n$ chessboard results in a high fraction of safe squares, showing it converges to $1/e^2$ as $n$ grows large.

Contribution

It demonstrates how to derive a simple closed-form limit for the fraction of safe squares using probabilistic algebraic techniques.

Findings

Probability of a high fraction of safe squares tends to $1/e^2$ as $n\to\infty$

Uses probabilistic tools like indicator variables, expectation, variance, and Stirling's formula

Provides a clear algebraic approach to a combinatorial chess problem.

Abstract

A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing $n$ queens on an $n\times n$ board. We examine related problems from placing $n$ rooks. We prove that as $n\to\infty$, the probability rapidly tends to 1 that the fraction of safe squares from a random placement converges to $1/e^2$. Our interest in the problem is showing how to view the involved algebra to obtain the simple, closed form limiting fraction. In particular, we see the power of many of the key concepts in probability: binary indicator variables, linearity of expectation, variances and covariances, Chebyshev's inequality, and Stirling's formula.