Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

May 29, 2007

Magic Squares

Do you have a lucky number? In ancient China, people believed that a special arrangement of nine numbers in a square was especially lucky. They engraved this pattern on stones or medallions that were worn as charms to ward off evil or bring good fortune.

Here's the pattern. Can you tell what's special about it?

Notice it contains all the numbers from 1 to 9. Better yet, the numbers in each row, column, and diagonal add up to 15.

Arrangements of numbers that add up to the same total in every row, column, and diagonal are known as magic squares. Melancholia, an engraving by the German artist Albrecht Dürer, includes a famous magic square. The rows, columns, and main diagonals all sum to 34.

The magic square is hanging on the wall to the upper right. Not only do the rows, columns, and diagonals total 34, so do the numbers in the corner squares and the numbers in the central four squares. Can you find other combinations within the square that add to 34? There are several. For example, If you divide the four-by-four square into four two-by-two squares, each of those squares will add up to 34. What's more, the numbers in the middle bottom squares read 1514, the year Dürer made the engraving.

Why pack so much number magic into one square? Astrologers in Dürer's time associated different types of magic squares with the planets, which, in turn, were thought to influence health. The brooding man is suffering from Saturn's "saturnine," or gloomy, influence. He hopes Jupiter's "jovial" four-by-four magic square will draw down, or decrease, Saturn's influence. (Of course this is all absolutely nutters, but that's the history of ideas for you.)

You don't have to use consecutive numbers or make sure that all the numbers are different. Here's a three-by-three example made up of only odd numbers.

Here's another one that consists of just prime numbers—numbers evenly divisible only by themselves and 1.

People have been looking for magic squares of various sorts for centuries. In colonial times, Benjamin Franklin used to make up magic squares when he got bored listening to political speeches. He's famous among number fanatics for an amazing eight-by-eight magic square containing the numbers from 1 to 64.

More recently, Lee Sallows, an electronics engineer in the Netherlands, discovered a magic square that has truly astonishing properties. Start with the following magic square:

Spell out the English words for each of the numbers:

Count the number of letters in each word, and enter the number in the appropriate space of a blank three-by-three grid:

The result is another magic square, which contains the consecutive numbers from 3 to 11! Such "alphamagic" squares can be found in other languages, too.

There's still a lot more to discover about magic squares, and there are many more to find. Maybe you'll encounter something interesting in searching for your own digital good-luck charm.

Muse, November/December 2003, p. 32-33.

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