A note on constructing and enumerating of magic squares

Résumé

Let $n\geq1$ be an integer. A magic square of order $n$ is a square table $n\times n$, say $A$, filled with distinct positive numbers $1,2,\ldots,n^2$ such that all cells of $A$ are distinct and the sum of the numbers in each row, column and diagonal is equal.
Let $M(n,s)$ be the set of all $n\times n$ $(0,1)$-matrices, say $T$, such that the number of $1$ in every row and every column of $T$ is $s$.
In this paper for every positive integer $k$ we find a new way for constructing magic squares of order $4k$. We show that the number of magic squares of order $4k$ is at least $|M(2k,k)|$. In particular we show that the number of magic squares of order $4k$ is at least $\frac{{2k \choose k}^2}{2}$.