Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence

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}$

Each of the lines through the center cell contain $3$ integers in arithmetic sequence.

Proof

By observation:

$\tuple {1, 5, 9}$: common difference $4$

$\tuple {2, 5, 8}$: common difference $3$

$\tuple {3, 5, 7}$: common difference $2$

$\tuple {4, 5, 6}$: common difference $1$

$\blacksquare$

Sources

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