Sums of Squares of Diagonals of Order 3 Magic Square

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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:

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


For the bottom-left to top-right diagonals:

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

$\blacksquare$


Also see


Sources