Lines through Center Square of Order 3 Magic Square are in Arithmetic Sequence
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Theorem
Consider the order 3 magic square:
- $\begin{array}{|c|c|c|}
\hline 2 & 7 & 6 \\ \hline 9 & 5 & 1 \\ \hline 4 & 3 & 8 \\ \hline \end{array}$
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): $9$