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.


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Proof

For the top-left to bottom-right diagonals:

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


For the bottom-left to top-right diagonals:

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

$\blacksquare$


Also see


Sources

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