Browsing by Author "Muurinen, Ismo"

Pierre de Fermat and Bernard Frénicle de Bessy discussed in their 1640 correspondence on magic squares. Frénicle did not appreciate Fermat’s contributions and it seems they have not been fully recognized even later. We will in this thesis look over their correspondence and study carefully every one of the ten "magic objects"by Fermat - nine squares and one cube. Through the ages, many methods have been developed to build magic squares, yet the one with which Fermat builds his even order magic squares, appears original even today. Fermat had related but slightly different method for odd order and even order squares. The idea behind odd order method was thoroughly explained later(without any reference to Fermat) by Frénicle as we point out here too. The even order method contains an original idea which we call "idea of self-supporting blocks". It is strongly based on the use of basic square as a starting point in construction process of magic square. After adopting this idea from Fermat, we use it first to provide an order 22 reconstruction for Fermat’s incomplete order 12 core of the full square. In the latter part of our work we show how this idea can be generalized for odd order squares as well. We demonstrate how it can be applied to build magic squares of any size, ordinary magic squares and bordered ones as well. Then the idea is applied to perfect magic cubes of orders divisible by 8. Frénicle presented in his letter as a challenge to Fermat a problem of magic squares with empty cells. It appears Fermat did not have time to respond to this challenge though he expressed he intended to. We will show how he might have done that. His method provides all the tools needed.