ELMO SL 2013 C1

Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$ , so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The magic square value of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$ . Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.)

Proposed by Ray Li