  A \emph{magic square} is an $n \times n$ array of distinct positive integers
whose sum along any row, column, or main diagonal is the same number. We
compute the number of such squares for $n=4$, as a function of either the magic
sum or an upper bound on the entries. The previous record for both functions
was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the
combination of hyperplane arrangements and Ehrhart's theory of lattice-point
enumeration.
