New Methods for Order n=4k+2 via Vertical Self-Complementarity and a Framework for Finding All Even-Order Magic Squares

Abstract

The discovery of analytical methods for constructing magic squares of order n=8t-2 is extremely rare. In this article, we present such a discovery. The magic squares generated have vertical symmetry and are numerous within the same order. The only drawback is that we were unable to discover a method for orders of the type n=8t+2. Most of the few existing methods for these orders require auxiliary magic squares, unlike ours, which is a straightforward method and can be better used in all branches of technology. Furthermore, these magic squares possess exceptional symmetry properties and, from the construction of one of them, many others (at least (n -2)!^n/2 ) of the same order are immediately obtained due to their vertical complementarities E = (e_{u, v} ) u, v ∈ I_n, e_{u, v} + e_u,n+1-v = n² + 1, for all u,v ∈ I_n ). We also present analytical and algebraic algorithms for constructing a large number of magic squares for all remaining even orders. The abundance of magic squares constructed by the algorithms led us to establish a project for the construction of all magic squares of all even orders. We have established a mathematical theory of the joint magicization of the diagonals of semimagic squares of even orders.