Question 103
Define an $n$-magic square to mean an $n \times n$ square matrix of non-negative integers such that the sum of all the entries in each row and each column is $m$ for some $m \in \mathbb{N}$. Also define an $n$-permutation matrix to mean an $n \times n$ square matrix of $n(n-1)$ zeroes and $n$ ones such that every row and every column of the array contains exactly one 1. Show that every $n$-magic square can be written as a sum of finitely many $n$-permutation matrices.