Magic Squares

Although originally of very little mathematical significance, magic squares were discovered by the Chinese emperor Yu the Great in approximately '2000 B.C' (Sanford, 1958, p74). As legend goes, Yu was presented two charts or diagrams by miraculous animals, during his reign of governing the empire. They were 'deemed to possess magical properties' (Rouse Ball, 1960, p119). 'The Ho Thu (diagram 8) was the gift of a dragon horse which came out of the Yellow River, and the Lo Shu (diagram 7) the gift of a turtle from the River Lo. The former (the 'River Diagram') was generally described as green, or in green writing, and the latter (the 'Lo River Writing') was traditionally red' (Needham, 1959, p56).

Diagrams 7 and 8 respectively, the Lo Shu Diagram and the Ho Thu Diagram.

The animals didn't simply hand the emperor these, Yu the Great actually found them physically on the creatures, for example the Lo Shu was on 'the back of the tortoise' (Sanford, 1958, p74).

They were considered to be magical as it was recognised that the sum of the numbers in every row, in every column, and in each major diagonal was the same. Clearly the Lo Shu is a straight forward magic square in which the integers added up along any column, row or major diagonal add up to 15 (see diagram 9). The Ho Thu is arranged a little different. Discarding the centre two integers (5 and 10) both the odd and even sets add up to 20 (see diagram 10). Note in diagrams 9 and 10 originally even numbers (yin numbers) were represented in black and odd (yang) in white.

Diagrams 9 and 10 respectively, modern day representations of the Lo Shu and Ho Thu.

It wasn't until the +13th century that the development of magic squares was introduced into the main current of mathematical thought. It was Emmanuel Moschopulus in 1460 who discovered the mathematical theory behind them. He recognised that 'if the integers be the consecutive numbers from 1 to n squared, the square is said to be of nth order, and in this case the numbers in any row, column, or diagonal is said to equal (1/2)n(n squared + 1)' (Ball, 1960, p119). Looking at diagram W we can see that this is true. It contains the numbers 1 to 9 (3 squared) and the sum of each line is 15 = (1/2)3(3x3+1). This is purely mathematical not magical.

Yang Hui (+13th century mathematician) was the first to actually study magic squares or vertical and horizontal diagrams as they were called. Some of the magic squares he made are very complicated. For example diagram 11 looks at the construction of a magic circle based on generally the same principals as that of magic squares. 'Here n concentric circles were cut by n diameters, and numbers placed on the points of intersection. The sum of the numbers on any diameter or along any circle was to be a constant' (Sanford, 1958, p74). In this example the sum along any diameter is equal to 147 and that of the circles is 138.

Diagram 11. Yang Huis magic circle.

He also gave some simple rules for their construction. 'For example, if the numbers from 1 to 16 are placed in an array of four rows and four columns, and the numbers at the corners of both the inner and outer square are transposed, a magic square will be produced in which all columns, rows and diagonals add up to 34' (Needham, 1959, p60) - see diagram 12.

Although considered today as of very little use, their significance lies primarily in interest and superstition. Many in the East still consider them good luck charms. Once upon a time engraved on a silver plate they were often prescribed as a charm against the plague and were used as decorations of fortune-telling bowls and amulets.

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