Sum of Cubes on Diagonals of Moessner's Order 4 Magic Square

Theorem

The sums of the cubes of the entries on the diagonals of Moessner's order $4$ magic square are equal.

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}$

\(\ds 12^3 + 3^3 + 14^3 + 5^3\)

\(=\)

\(\ds 1728 + 27 + 2744 + 125\)

\(\ds \)

\(=\)

\(\ds 4624\)

\(\ds 9^3 + 2^3 + 15^3 + 8^3\)

\(=\)

\(\ds 729 + 8 + 3375 + 512\)

\(\ds \)

\(=\)

\(\ds 4624\)

$\blacksquare$

Sources

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