# Magic Squares from Primary Classroom to Postgraduate Research in 10 Simple Exercises: Tim Roberts Invites Students and Their Teachers to Engage with the Mathematics of Magic Squares

Roberts, Tim, Australian Primary Mathematics Classroom

It is not often that one can introduce primary school students to a problem at the forefront of mathematics research, and have any expectation of understanding; but with magic squares, one can do exactly that. Magic squares are an ideal tool for the effective illustration of many mathematical concepts. This paper assumes little prior knowledge on the part of the student except for addition and multiplication, reflection and rotation; but, using questions and exercises throughout, finishes by posing a problem tough enough to test postgraduate mathematics students, to which no-one has yet managed to find a solution.

Who knows what a magic square is?

Some students may know: a square consisting of numbers, where the total of each row and each column is the same. Close! In fact, to be truly regarded as a magic square, each of the diagonals should sum to that same total, too. If they do not, the square is often called semi-magic.

All of the numbers used are generally required to be different. If this is not enforced, then we could have a magic square in which every entry was 1, for example--which tends to be rather boring (see Figure 1).

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This article will show how magic squares can be used to illustrate many mathematical concepts.

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Magic squares

Exercise 1. Draw a 3 by 3 grid. Without any clues, see if students can fill in the numbers 1 to 9 so that the resulting square is magic.

Some students will be able to find a solution. If not, you can give an additional hint: put the number 5 in the middle. Figure 2 shows one solution.

It should be noted that every row, and every column, and even the two main diagonals, sums to the same total; in this case, 15. This is called the magic constant.

Exercise 2. Rotate the square through 90 degrees. Is it still magic?

Exercise 3. Reflect the square vertically or horizontally. Is it still magic?

The answer to both questions is: yes. Figure 3 shows the same square rotated through 90 degrees and Figure 4 shows it again reflected about a vertical axis.

Exercise 4. Is it possible to construct another 3 by 3 magic square using the numbers 1 through 9 where the magic constant is any other value than 15?

No: since all of the numbers 1 through 9 sum to 45, each row and each column must be 45 / 3, or 15. It has also been shown that 5 must always go in the centre square.

Larger magic squares are possible. A 4 by 4 magic square is illustrated in Figure 5. Of course, all rows and columns, as well as both diagonals, sums to the magic constant, which is the sum of the numbers 1 through 16, divided by 4, which equals 34.

Arithmetic and 3 by 3 squares

Looking at bigger squares can be very interesting, but for our purposes, let us concentrate on 3 by 3 squares.

Exercise 5. Try doubling every entry in the square. Is the square still magic?

Yes. Of course, the magic constant is doubled too, because we are no longer using the numbers 1 to 9, but rather, 2 to 18 (see Figure 6).

Exercise 6. Add a number of your choice--say 25--to every entry. Is the square still magic?

Yes again. The magic constant is whatever it was before, plus three times the number we have just added (see Figure 7).

Exercise 7. Take our original square (using the numbers 1 to 9) and square each entry; that is, multiply it by itself. Is the square still magic?

No. Unfortunately we end up with a square which is not magic at all! (see Figures 8 and 9).

So, we can multiply every entry, or add a particular number to each entry, and in both cases the square remains magic; but we cannot multiply each entry by itself! …

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