Uniform Step Magic Squares Revisited

Abstract

We derive an alternative rule for generating uniform step magic squares. The compatibility conditions for the proposed rule are simpler than the analogous conditions for the classical uniform step rule. We exploit this fact to enumerate all uniform-step magic squares of every given odd order. Our main result states that if \(p = \prod_{i=1}^l q_i^{r_i}\) is the prime factorization of a positive odd number \(p\), then there exist \(\kappa(p) =\prod _{i=1}^l \kappa(q_i^{r_i})\) uniform step magic squares of order \(p\), where
\(\kappa (q_i^{r_i})=[\tau (q_i^{r_i})]^2-\lambda (q_i^{r_i}),\lambda(q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})^2[2(q_i^{2r_i-1}+1)^2/(q_i+1)^2+q_i^{3r_i-1}(q_i^{r_i}-3q_i^{r_i-1})]\) and \(\tau (q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})(q_i^{2r_i+1}-2q_i^{2r_i}-q_i^{2r_i-1}+2)/(q_i+1)\) for \(i=1,\ldots,l\)