Bordered magic squares: Elements for a comprehensive approach

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 like 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 caracterise 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.