Definition:Multiplicative Magic Square
Definition
A multiplicative magic square is an arrangement of $n^2$ distinct numbers into an $n \times n$ square array such that:
- the product of the elements of each row
- the product of the elements in each column
- the product of the elements along each diagonal
are the same.
Order
An $n \times n$ multiplicative magic square is called an order $n$ multiplicative magic square.
Magic Constant
The magic constant of a multiplicative magic square is the number that each of the rows and columns multiplies up to.
Examples
Order $1$
The Order $1$ multiplicative magic square is trivial:
- $\begin{array}{|c|}
\hline 1 \\ \hline \end{array}$
Order $3$
Order $3$ Multiplicative Magic Square with Smallest Magic Constant
The order $3$ multiplicative magic square with the smallest magic constant is as follows:
- $\begin{array}{|c|c|c|}
\hline 18 & 1 & 12 \\ \hline 4 & 6 & 9 \\ \hline 3 & 36 & 2 \\ \hline \end{array}$
Its magic constant is $216$.
Order $4$
Order $4$ Magic Square with Smallest Magic Constant
The order $4$ multiplicative magic square with the smallest magic constant is as follows:
- $\begin{array}{|c|c|c|c|}
\hline 1 & 15 & 24 & 14 \\ \hline 12 & 28 & 3 & 5 \\ \hline 21 & 6 & 10 & 4 \\ \hline 20 & 2 & 7 & 18 \\ \hline \end{array}$
Its magic constant is $5040$.
Also known as
Some sources call this a multiplication magic square.
Also see
- Results about multiplicative magic squares can be found here.
Sources
- Weisstein, Eric W. "Multiplication Magic Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultiplicationMagicSquare.html