Sums of Squares of Lines 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 rows, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same rows of that same order 3 magic square when reflected in a vertical axis:
$\begin{array}{|c|c|c|}

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


Similarly:

The sums of the squares of the columns, when expressed as $3$-digit decimal numbers, are equal to the sums of the squares of those same columns of that same order 3 magic square when reflected in a horizontal axis:
$\begin{array}{|c|c|c|}

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


Proof

For the rows:

\(\ds 276^2 + 951^2 + 438^2\) \(=\) \(\ds 76176 + 904401 + 191844\)
\(\ds \) \(=\) \(\ds 1172421\)
\(\ds 672^2 + 159^2 + 834^2\) \(=\) \(\ds 451584 + 25281 + 695556\)
\(\ds \) \(=\) \(\ds 1172421\)


For the columns:

\(\ds 492^2 + 357^2 + 816^2\) \(=\) \(\ds 242064 + 127449 + 665856\)
\(\ds \) \(=\) \(\ds 1035369\)
\(\ds 294^2 + 753^2 + 618^2\) \(=\) \(\ds 86436 + 567009 + 381924\)
\(\ds \) \(=\) \(\ds 1035369\)

$\blacksquare$


Also see


Sources

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