Magic Squares

By Ian Stewart

According to a Chinese myth, the Emperor Yu, who lived in the third millenium BC, came across a sacred turtle in a tributary of the Yellow River, with strange markings on its shell. These markings are now known as the Lo Shu ("Lo river writing").

The markings are numbers, and they form a square pattern:

4 9 2
3 5 7
8 1 6

Here every row, every colum and every diagonal adds to the same number, 15. A number square with these properties is said to be magic, and the number concerned is its magic constant. Usually the square is made from successive whole numbers, 1, 2, 3, 4, and so on, but sometimes this condition is relaxed.

In 1514 the artist Albrecht Durer produced an engraving "Melancholia," containing a 4x4 square. The middle numbers in the bottom row are 15-14, the date of the work. This square contains the numbers

16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1

and has magic constant 34.

Using consecutive whole numbers 1, 2, 3,...., and counting rotations and reflections of a given square as being the same, there are precisely:

. 1 magic square size 3 x 3
. 880 magic squares of size 4 x 4
. 275,305, 224 magic squares size 5 x 5

The number of 6 x 6 magic squares is unknown, but has been estimated by statistical methods to be about 1.77 x 10 (19)



From the book Professor Stewart’s Cabinet of Mathematical Curiosities by Ian Stewart. Excerpted by arrangement with Basic Books, a member of the Perseus Books Group. Copyright © 2009.