The Evolution of Magic Squares in China

Abstract

MAGIC SQUARES are diagrams composed of numbers or letters, specially arranged in a rectangular pattern to produce certain distinctive effects. Although the lettered squares were very common in the Islamic world, they do not appear in China, so we shall here consider only the numbered ones. These numerical magic squares were normally composed of all the numbers from one to the square of the base number-for example, in the magic square of three, this would include the numerals from 1 to 32, or 9-and these are all carefully placed so that every row and every column, and the two main diagonals, will each add up to the same number. This common sum is usually called the Constant. Although these numerical magic squares are now usually relegated to that minor subdivision of Western Science known as recreations, people of past centuries, all over the civilized world, took them very seriously. Their construction was regarded as a worthwhile study in mathematical techniques, or even as a creative artistic expression (in cultures with strong aniconic traditions); and the finished squares were considered as solemn religious symbols, or as charms and talismans with innate powers for good or evil. As magical diagrams they played an important part in the occult studies which formed a branch of mediaeval proto-science among the Chinese, the Hindus, and the Arabs, and ultimately among the Europeans of the Early Renaissance. And, in working with them, men made discoveries regarding the relationships of numbers which led to significant developments in Mathematics.'