Sum of Squares on Pairs of Rows and Columns of Moessner's Order 4 Magic Square

Theorem

The sums of the squares of the entries are equal on the following pairs of rows and columns of Moessner's order $4$ magic square:

Rows $1$ and $4$

Rows $2$ and $3$

Columns $1$ and $4$

Columns $2$ and $3$.

Proof

Recall Moessner's order $4$ magic square:

$\begin{array}{|c|c|c|c|}

\hline 12 & 13 & 1 & 8 \\ \hline 6 & 3 & 15 & 10 \\ \hline 7 & 2 & 14 & 11 \\ \hline 9 & 16 & 4 & 5 \\ \hline \end{array}$

Rows $1$ and $4$

\(\ds 12^2 + 13^2 + 1^2 + 8^2\)

\(=\)

\(\ds 144 + 169 + 1 + 64\)

\(\ds \)

\(=\)

\(\ds 378\)

\(\ds 9^2 + 16^2 + 4^2 + 5^2\)

\(=\)

\(\ds 81 + 256 + 16 + 25\)

\(\ds \)

\(=\)

\(\ds 378\)

$\Box$

Rows $2$ and $3$

\(\ds 6^2 + 3^2 + 15^2 + 10^2\)

\(=\)

\(\ds 36 + 9 + 225 + 100\)

\(\ds \)

\(=\)

\(\ds 370\)

\(\ds 7^2 + 2^2 + 14^2 + 11^2\)

\(=\)

\(\ds 49 + 4 + 196 + 121\)

\(\ds \)

\(=\)

\(\ds 370\)

$\Box$

Columns $1$ and $4$

\(\ds 12^2 + 6^2 + 7^2 + 9^2\)

\(=\)

\(\ds 144 + 36 + 49 + 81\)

\(\ds \)

\(=\)

\(\ds 310\)

\(\ds 8^2 + 10^2 + 11^2 + 5^2\)

\(=\)

\(\ds 64 + 100 + 121 + 25\)

\(\ds \)

\(=\)

\(\ds 310\)

$\Box$

Columns $2$ and $3$

\(\ds 13^2 + 3^2 + 2^2 + 16^2\)

\(=\)

\(\ds 169 + 9 + 4 + 256\)

\(\ds \)

\(=\)

\(\ds 438\)

\(\ds 1^2 + 15^2 + 14^2 + 4^2\)

\(=\)

\(\ds 1 + 225 + 196 + 16\)

\(\ds \)

\(=\)

\(\ds 438\)

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16$