Magic squares cornered

摘要

Dame Kathleen Ollerenshaw, one of England's national treasures, has solved a long-standing, extremely difficult problem involving the construction and enumeration of a certain type of magic square. The solution comes in a book written with David Bree. First, some background on magic squares, and their hierarchy of perfection. For many centuries, mathematicians — especially those concerned with combinatorics — have been challenged by magic squares. These are arrangements of n~2 distinct integers in an n x n array such that each row, column and main diagonal has the same sum. The sum is called the magic constant, and n is called the square's order. Traditional magic squares are made with consecutive integers starting with 0 or 1. If it starts with 0 it can be changed to a square starting with 1 simply by adding 1 to each cell.