magic square

A magic square of order $n$ is an $n\times n$ array using each one of the numbers $1,2,3,\mathrm{\dots },n^2$ once and such that the sum of the numbers in each row, column or main diagonal is the same.

Example:

It’s easy to prove that the sum is always $\frac{1}{2}n(n^2+1)$. So in the example with $n=3$ the sum is always $\frac{1}{2}(3\times 10)=15$.

One way to generalize this concept is to allow any numbers in the entries, instead of $1,2,\mathrm{\dots },n$.