Properties of Moessner's Order 4 Magic Square

Properties of Moessner's Order $4$ Magic Square

Recall Alfred Moessner's order $4$ magic square:

$\begin{array}{|c|c|c|c|}

\hline 12 & 13 & 1 & 8 \\ \hline 6 & 3 & 15 & 10 \\ \hline 7 & 2 & 14 & 11 \\ \hline 9 & 16 & 4 & 5 \\ \hline \end{array}$

This has the following properties:

Sum of Cubes on Diagonals

The sums of the cubes of the entries on the diagonals of Moessner's order $4$ magic square are equal.

Sum of Squares on Pairs of Rows and Columns

The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square:

Rows $1$ and $4$

Rows $2$ and $3$

Columns $1$ and $4$

Columns $2$ and $3$.