Smallest Multiplicative Magic Square is of Order 3
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Theorem
The order of the smallest multiplicative magic square is $3$, for example:
- $\begin{array}{|c|c|c|} \hline 18 & 1 & 12 \\ \hline 4 & 6 & 9 \\ \hline 3 & 36 & 2 \\ \hline \end{array}$
Its magic constant is $216$.
Proof
Suppose there were an order $2$ multiplicative magic square $M$.
Let $a$ be the element of row $1$ and column $1$.
Let $a b$ be the magic constant of $M$.
Then $b$ is:
- the element of row $1$ and column $2$, to make the product of row $1$ equal to $a b$
- the element of row $2$ and column $1$, to make the product of column $1$ equal to $a b$
and so on.
But it is a principle of multiplicative magic squares that the elements are all distinct.
Hence no order $2$ multiplicative magic square can exist.
$\blacksquare$