Generating and enumerating magic squares

Abstract

Magic squares have long provided a fruitful source of classroom examples and stimulating exercises in recreational mathematics. Attention is often devoted to the so-called normal magic squares of a given order n in which the entries are the integers from 1 to n2. In these the size must be at least 4 for examples to be sufficiently plentiful: there is essentially only one normal magic square of order 3 (apart from rotations and reflections), while there are 880 of order 4 and several millions of order 5. However, if we drop the requirement of normality, the 3 × 3 square becomes a surprisingly rich subject for investigation, which can challenge pupils at all levels. In this note we discuss the following problems: (1) find a formula to generate all 3 × 3 magic squares with a given magic number m (the sum of entries in any row, column or diagonal), with no restriction on the entries—they can be any real numbers; (2) find conditions under which the square has all entries positive integers; (3) determine how many essentially distinct positive integer squares there are for a given m.