# Sums of Squares of Diagonals of Order 3 Magic Square

## Theorem

Consider the order 3 magic square:

$\begin{array}{|c|c|c|} \hline 2 & 7 & 6 \\ \hline 9 & 5 & 1 \\ \hline 4 & 3 & 8 \\ \hline \end{array}$

The sums of the squares of the diagonals, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same diagonals of that same order 3 magic square when reversed.

## Proof

For the top-left to bottom-right diagonals:

 $\displaystyle 258^2 + 714^2 + 693^2$ $=$ $\displaystyle 66564 + 509796 + 480249$ $\displaystyle$ $=$ $\displaystyle 1056609$ $\displaystyle 852^2 + 417^2 + 396^2$ $=$ $\displaystyle 725904 + 173889 + 156816$ $\displaystyle$ $=$ $\displaystyle 1056609$

For the bottom-left to top-right diagonals:

 $\displaystyle 456^2 + 312^2 + 897^2$ $=$ $\displaystyle 207936 + 97344 + 804609$ $\displaystyle$ $=$ $\displaystyle 1109889$ $\displaystyle 654^2 + 213^2 + 798^2$ $=$ $\displaystyle 427716 + 45369 + 636804$ $\displaystyle$ $=$ $\displaystyle 1109889$

$\blacksquare$