Constructing Magic Squares: an integer constraint satisfaction problem and a fast approach

Magic squares are a fascinating mathematical challenge that has intrigued mathematicians for centuries. Given a positive (and possibly large) integer $n$, one of the main challenges that still remains is to find, within a computational time, a magic square of order $n$, that is, a square matrix of order $n$ with unique integers from $a_{\min }$ to $a_{\max }$, such that the sum of each row, column, and diagonal equals a constant $\mathcal{C}(A)$. In this work, we first present an integer constraint satisfaction problem for constructing a magic square of order $n$. Nonetheless, the solution time of this problem grows exponentially as the order increases. To overcome this limitation, we also propose a that constructs magic squares depending on whether $n$ is odd, singly even, or doubly even. Moreover, we provide a proof of the correctness of this novel approach. Our numerical results show that our method can construct magic squares of order up to \num{70000} in less than \num{140} seconds, demonstrating its efficiency and scalability.