Worksheet Number Twelve

Magic Squares

In our culture, the art of constructing magic squares is a type of recreational mathematics. For the ancient Chinese, there was a mystical significance to magic squares. The oldest known magic square, the Luo Shu, originally came to the legendary emperor Yu, who had magical powers to control floods. He got the Luo Shu from the back of a turtle which crawled out of the Luo River and presented the magic square to him in the pattern of dots on its shell. History dates magic squares, and the Luo Shu in particular, to the Zhou Dynasty (1027-256 B.C.), in the warring states period of 475-221 B.C.

4 9 2

3 5 7 This is the Luo Shu.

8 1 6

In ancient China, balance was viewed as having Yang, positive energy, in harmony with Yin, negative energy, the resulting stability being Yinyang. The number 5 represented the emperor, China, the center of everything. It also stood for the five elements from which everything is made: fire, earth, metal, water and wood. Notice the centering 5 in the Luo Shu, and that it is the average, or mean, of any two numbers which are opposite around the 5. Yang numbers are odd and Yin numbers are even. They are arranged in pairs around the center 5.

Find some other (3X3) magic squares, using the Luo Shu any way you choose. By magic square I mean 9 numbers, which need not be 1 through 9, arranged so that the sum along any row, column or diagonal is the same ”magic” number.

Now let’s ascend to (4X4) magic squares. Here is a starting point, a magic square used in India to protect travelers.

4 9 5 16

14 7 11 2 Here the magic number is what?

15 6 10 3

1 12 8 13

Can you use ideas from the (3X3) problems to make any more (4X4) magic squares?

A magic square with magic number 84 was used to soothe a crying child. Can you do mothers of infants in this country a favor?