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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$