by Claudia Zaslavsky
How can we help our students to develop those important skills with numbers and computation, without boring them with drillnkill worksheets? One answer is magic squares. The construction and analysis of magic squares provides practice in mental arithmetic, operations with numbers, geometry, and measurement, and encourages logical reasoning and creativity, all in a game-like setting. Each student can proceed at his or her own pace. Best of all, the students correct their own work; finding errors is part of the process.
Magic squares are popular in many cultures. The Chinese may have invented them. According to their legend, the emperor and his court were sailing down the River Lo about 4,000 years ago. Suddenly a turtle appeared out of the water. On the turtles back was a design with nine numbers, expressed in combinations of dots. Each number, from 1 to 9, appeared exactly once. They called this design of numbers the Lo Shu, or river map.
Why is this square of numbers magic?
The number 15 is the magic sum for this magic square.
The art of making magic squares spread to other landsto India and Japan, to the Middle East, to Africa, and finally to Europe and America. In the Middle East and Africa magic squares became part of Islam, the religion of the Muslims.
Muslim scholars enjoyed making large, complicated magic squares. They called this the science of secrets. Very often they would introduce mistakes on purpose, to keep ordinary people from learning the science. Perhaps these squares were a code to send secret messages! In the 18th century the Nigerian scholar Muhammad ibn Muhammad wrote a whole book about magic squares in the Arabic language. 1
Students can start off by completing magic squares in which several numbers have already been entered in their correct positions. Look at the three examples below.
In all these squares, the numbers from one to nine occur just once each, and the magic sum is 15. This exercise hones skills in addition and subtraction with small numbers. Encourage children to carry out the computations mentally. Younger children may require more helpperhaps five or six given numbers in each square, rather than three. Analysis of magic squares Ask the students to examine the three squares they have completed.
Once students have completed their analysis, encourage them to construct their own three-by-three magic squares, using the same rules. There are eight such squares altogether. How many can they find? It may be easier for the children to manipulate disks or cardboard squares numbered from one to nine, rather than making holes in the paper as they erase their errors. Some of my students were motivated to draw large, neat squares and color them appropriatelygood bulletin board material, as well as an exercise in geometry and measurement.
This exercise should encourage the students to think about the properties of numbers. Is the sum of two odd numbers odd or even? Can they add two odd numbers and an even number to obtain a sum of 15? Why not place odd numbers in some or all of the corners of the magic square? Can the center number be different from 5? Students might explore these questions and note the outcomes. Different groups of students can be assigned various alternatives for exploration, and then explain to the class why they dont work. This is a good introduction to algebraic thinking and generalization.
Transformations of magic squares
By now your students may have discovered the eight possible three-by-three magic squares using the numbers 1 to 9. Actually they can make all eight magic squares by drawing just one square in a special way2.
Each student will draw a large magic square similar to the Lo Shu, the Chinese magic square. They will need two magic markers of different colors, lets say red and black. They will copy the pattern of dots in the figure below, coloring the even numbers red and the odd numbers black.
Write the word Bottom at the bottom of the sheet of paper. Students should place scrap paper under the square to keep the desk clean.
Then turn over the paper. If the lines and dots dont show through clearly, go over them with the markers. Now each student has two magic squares, one on each side of the paper. Side One has the word Bottom marked on it. Write Side One at the top. Turn over the sheet and write Side Two at the top. Can the students tell, without counting, the total number of dots in the magic square? [9 X 5 = 45 dots]
Students will need another sheet of paper as a worksheet on which to record the outcomes when they rotate (turn) and reflect (flip) the magic square in different ways.
ROTATIONS: Ask the students to rotate Side One a quarter turn, or 90 degrees, clockwise, and record the corner numbers in a square on the worksheet. Then they rotate it another quarter turn, and another, recording the results each time, until they come back to the starting position. They should have recorded four different magic squares.
REFLECTIONS: Next they will flip the magic square in different ways. Each time they will start with Side One (as in Figure 4) and record the corner numbers:
Building on magic squares
There is no limit to the level of computational skill involved in working with magic squares. Suppose you use the consecutive numbers from 3 to 11. How will the center number and the magic sum change? Can you multiply each number in the basic square by some quantity, a whole number or a fraction, and still have a magic square? Can you use a three-by-three square of numbers from a calendar page to make a magic square? These are some of the questions students may explore in the activity Playing with Magic Squares 3.
PLAYING WITH MAGIC SQUARES
In the last activity you made 3 X 3 magic squares with the numbers 1 to 9. The center number was 5. The magic sum was 3 X 5, or 15. The even numbers were in the corners of the square.
Suppose you want to make a magic square using the numbers 2 to 10. All you have to do is add a 1 to each number in the magic square you already made. Before you do that, figure out what the magic sum should be. If you add a 1 to each number in a row, the magic sum should be 15 + 3, or 18. The center number is even and the numbers in the corners are odd.
CHANGING MAGIC SQUARES
Sheet of paper. If you have centimeter or half-inch graph paper, that would be helpful. Lined paper is also good.
MAKING THE MAGIC SQUARES
Things to think about and do
Four-by-four magic squares
Thus far we have discussed only three-by-three magic squares. Four-by-four squares provide another level of fascination. The artist Albrecht Dürer included such a square in his painting Melancholia I. In the bottom row were the numbers 15 and 14, to indicate that he painted the picture in the year 1514. The square below is called Panmagic or diabolical, due to the many ways in which four numbers can be added to attain the sum 34. The four numbers, when connected in order, form geometric shapessquares of different sizes, parallelograms, triangles, and trapezoids.4
According to the National Council of Teachers of Mathematics document, Principles and Standards for School Mathematics, published in 2000: Mathematics is one of the greatest cultural and intellectual achievements of humankind, and citizens should develop an appreciation and understanding of that great achievement, including its aesthetic and recreational aspects (page 4). Activities with magic squares fulfill several of the specific standards discussed in this document.
On the other hand, students say that work with magic squares is too much fun to be real math!
Activities reprinted from More Math Games & Activities from Around the World by Claudia Zaslavsky, published by Chicago Review Press, distributed by Independent Publishers Group. It may be ordered by calling 800-888-4741 or visiting http://www.ipgbook.com
Copyright 2003 by Synergy Learning International, Inc. All rights reserved.
- Claudia is a retired teacher of mathematics and author of thirteen books and many articles dealing with multicultural perspectives and equity issues in mathematics education. (2003) http://www.math.binghamton.edu/zaslav/cz.html