When Rooks Miss: Probability through Chess

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 $n\to \mathrm{\infty }$, the probability rapidly tends to 1 that the fraction of safe squares from a random placement converges to $1\mathrm{/}{e}^2$. 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.