Reprinted from Connect
Vol.17 No.2, November/December, 2003
Focus on: Puzzles and Games
Grade Level: 3-5
Grade Level: 6-8

Magic Squares

by Claudia Zaslavsky

How can we help our students to develop those important skills with numbers and computation, without boring them with drill’n’kill 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 turtle’s 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.”

• Add the dots in each row across. The sum is 15.
• Add each column going down. The sum is 15.
• Add each diagonal. The sum is 15.

The number 15 is the magic sum for this magic square.

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.

• What number is in the center space? [5]
• How is the center number related to the first and last numbers? [average of 1 and 9]
• How is the magic sum related to the center number? [15 = 3 X 5]
• What type of numbers are in the corners? [even numbers]
• What numbers are in the diagonals? [one diagonal has consecutive numbers]

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 appropriately—good 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 don’t 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, let’s 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.

Then turn over the paper. If the lines and dots don’t 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:

• Flip so that the row 3-5-7 remains in the same position. This row is the line of symmetry.
• Flip so that the column 9-5-1 remains in the same position.
• Flip so that the diagonal 4-5-6 remains in the same position.
• Flip so that the diagonal 2-5-8 remains in the same position.

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

MATERIALS:

Sheet of paper. If you have centimeter or half-inch graph paper, that would be helpful. Lined paper is also good.
Pencil
Eraser
Ruler

MAKING THE MAGIC SQUARES

1. Draw several squares, divided into 9 small squares.
2. Copy a magic square in the previous activity, or use this basic magic square:









3. Make a new magic square using the numbers 2 to 10. What is the magic sum? What is the center number?
4. Make another magic square using the even numbers from 2 to 18. Before you make the square, figure out what number to put in the center. What magic sum do you expect?

Things to think about and do

1. Suppose you take the basic magic square and multiply each number by 3. What is the magic sum? What number goes in the center? What is the sum of all the numbers? Check your work by making the magic square.
2. Suppose you add 5 to each number in the basic magic square. What is the center number? What is the magic sum? What is the sum of all the numbers? Make the magic square and check your answers.
3. Choose any page of a monthly calendar. Draw a square around a 3 X 3 block of nine numbers.
   

   
   

   
   
   
   
   Rearrange these nine numbers to form a magic square. Here is a hint: the center number remains the same. If it is even, the corner numbers are odd; if it is odd, the corner numbers are even.
   What is the sum of all the numbers? How is it related to the center number?
4. Muslim scholars purposely wrote magic squares with mistakes. Find the mistake in this magic square. The magic sum is 30.

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 shapes—squares of different sizes, parallelograms, triangles, and trapezoids.⁴

On the other hand, students say that work with magic squares is too much fun to be real math!

Notes
1. Zaslavsky, Claudia. Africa Counts: Number and Pattern in African Cultures, 3rd edition. Chicago: Lawrence Hill Books, 1999.
2.———————. Math Games and Activities from Around the World. Chicago: Chicago Review Press, 1998.
3, 4.———————. More Math Games and Activities from Around the World. Chicago: Chicago Review Press, 2003.

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 Zaslavsky - 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
List all articles by Claudia Zaslavsky