Two thousand years of numerical magic squares

Abstract

A magic square is any square-shaped array of numbers or letters exhibiting certain properties: in numerical squares it is the constant sum of numbers in each row, column, or diagonal. This is the constant of a square, while the number of cells in a row gives its order. Any set of numbers can be used if the condition of a constant summation is met. In natural magic squares only natural numbers from 1 to 2 are used. The same set of natural numbers written sequentially into a square grid leads to a natural square which is not magic. Numerical magic squares attracted attention in the distant past. They have been used in charms, talismans, and in other objects destined to magical use. More rarely, these squares appeared in philosophical or alchemical speculations as expressions of certain ideas, or even as an instruction to experiments. Now studied as problems in number theory, numerical magic squares once had deeper significance, far from purely mathematical questions.