# Magic Squares

By de Mestre, Neville | Australian Mathematics Teacher, Fall 2013 | Go to article overview

# Magic Squares

de Mestre, Neville, Australian Mathematics Teacher

The following is the basis of an extension topic that I recently gave to mathematically-advanced Year 7 students at a local primary school. It could also be used for higher classes.

Consider the following 3 x 3 square array of numbers.

The total for each row, each column and each of the two diagonals is 30. Such square arrays are termed magic squares. Here is an easy one for your students to complete.

The next one is a little harder.

The problem here is that the total is not known. Some of your students may get to the answer by trial and error. At least they will find out that not all 3 x 3 square arrays are magic. Others may like to use algebra by letting one of the unknown corner numbers be z, say.

Next ask your students to make up a magic square using the numbers 1, 1, 1, 2, 2, 2, 3, 3, 3. After a short time ask them what total are they aiming for?

Some will guess correctly, but you really want someone to tell you that the required total can be determined mathematically as the sum of all the numbers divided by 3. Now they should try to find the magic square, and many will get the row and column totals correct, but not both the diagonals. The correct answer must have the three 2s along a diagonal. The reason for this is that since 2 + 2 + 2 = 6 (the required total), we cannot form two other triplets from the remaining six numbers which each add to 6. Therefore 2. 2, 2 cannot be in a row or column. The solution can be obtained by pencil and paper, but it is more productive to use a hands-on approach. Write the numbers on nine slips of paper, and then arrange them in three groups each adding to six. The simplest are three lots of 1 + 2 + 3. Place them in three rows to form the 3 x 3 array. Now move them within their rows so that the columns each add to 6. Finally check the sums of both diagonals and, if they do not come to 6, swap whole rows or swap whole columns until they do.

Next, ask them to step back a little and make up a 2 x 2 magic square. They should soon discover that the only ones possible have the same number in each cell.

Here are some other groups of nine numbers that your students could try to place in 3 x 3 magic squares.

1. 1, 2, 3, 4, 5, 6, 7, 8, 9

2 1/24, 1/12, 1/8, 1/6, 5/24, 1/4, 7/24, 1/3, 3/8

3. 7, 37, 43, 67, 73, 79, 103, 109, 139

4. x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4

The hands-on approach is suggested for these. …

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