Magic Constant of Smallest Prime Magic Square
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Theorem
The magic constant of the smallest prime magic square is $111$.
Proof
The smallest prime magic square (including $1$) is:
- $\begin{array}{|c|c|c|}
\hline 67 & 1 & 43 \\ \hline 13 & 37 & 61 \\ \hline 31 & 73 & 7 \\ \hline \end{array}$
As can be seen by inspection, the sums of the elements in the rows, columns and diagonals are $111$:
| \(\ds 67 + 1 + 43\) | \(=\) | \(\ds 111\) | ||||||||||||
| \(\ds 13 + 37 + 61\) | \(=\) | \(\ds 111\) | ||||||||||||
| \(\ds 31 + 73 + 7\) | \(=\) | \(\ds 111\) |
| \(\ds 67 + 13 + 31\) | \(=\) | \(\ds 111\) | ||||||||||||
| \(\ds 1 + 37 + 73\) | \(=\) | \(\ds 111\) | ||||||||||||
| \(\ds 43 + 61 + 7\) | \(=\) | \(\ds 111\) |
| \(\ds 67 + 37 + 7\) | \(=\) | \(\ds 111\) | ||||||||||||
| \(\ds 43 + 37 + 31\) | \(=\) | \(\ds 111\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $111$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $111$