Bordered Magic Squares: Elements for a comprehensive approach

Resumen

General methods for the construction of magic squares of any order have been searched for centuries. Several 'standard strategies' have been found for this purpose, such as the 'knight movement', or the construction of bordered magic squares, which played an important role in the development of general methods. What we try to do here is to give a general and comprehensive approach to the construction of magic borders, capable of assuming methods produced in the past as particular cases. This general approach consists of a transformation of the problem of constructing magic borders to a simpler - almost trivial - form. In the first section, we give some definitions and notation. The second section consists of the exposition and proof of our method for the different cases that appear (Theorems 1 and 2). As an application of this method, in the third section we caracterize magic borders of even order, giving therefore a first general result for bordered magic squares. Although methods for the construction of bordered magic squares have always been presented as individual succesful attempts to solve the problem, we will see that a common pattern underlies the fundamental mechanisms that lead to the construction of such squares. This approach provides techniques for constructing many magic bordered squares of any order, which is a first step to construct all of them,and finally know how many bordered squares are for any order. These may be the first elements of a general theory on bordered magic squares.