Negative Magic Squares
AO elaboration and other teaching resources
Use addition with negative numbers
Know the idea of, and be able to construct, magic squares
Devise and use problem solving strategies to explore situations mathematically (be systematic).
First of all, if the class hasn’t heard of magic squares, then you may need to tell them that a magic square is an arrangement like the one below where the vertical, horizontal and diagonal lines of numbers all add up to the same value. This ‘same value’ is called the sum of the magic square.
Magic squares are interesting objects in both mathematics proper and in recreational mathematics. So they are objects that students should have heard about and experienced. The problems in this sequence give students the opportunity to use the new numerical or algebraic concepts that they will have acquired at that Level, along with magic squares.
It’s a critical part of this and some later problems that three times the centre square is equal to the sum of the magic square. We’ll prove this in the Extension to this lesson and in the Level 5 lesson (The Magic Square).
This problem is in the sequence of problems on magic squares. The first of these is A Square of Circles (at Level 2), and no attempt is made to actually explore magic square properties there. The second lesson is Little Magic Squares (Level 2). There are essentially two magic square problems at Level 3 – Big Magic Squares and Decimal Magic Squares.
As well as this lesson, at Level 4, Fractional Magic Squares uses fractions. This is followed by The Magic Square, Level 5. Finally, Difference Magic Squares at Level 6, looks at an interesting variation of the magic square concept.
Tui has begun to really like magic squares. She decided to make all of the magic squares that she could using the numbers –2, 4 and 10 down the main diagonal. How many can she make?
It took her quite a while because she didn’t know that the sum of a magic square was always three times the number in the centre.
- Talk about square ‘arrays’ of numbers like the ones in A Square of Circles. Ask the class if you can put numbers into these arrays so that the rows have the same sum; the columns have the same sum; all of the rows, columns and diagonals have the same sum.
- Show them a magic square such as the one below.
- Get them to check that the rows all have the same sum (of 12); that the columns all have the same sum; and that the diagonals have the same sum.
- Tell them that these things are called magic squares and that the sum of a magic square is the common sum of the rows, columns and diagonals.
- Tell them Tui’s problem.
- Ask them to go away in pairs and see how many magic squares they can find.
- Get some of the pairs to report back. Can they prove that the arrays they have produced are magic squares?
- How many different magic squares can they find? How many do they think there are?
- It is probable that they will begin to see that there are an infinite number. They may think that for every different number you put in the middle top row entry, you get a different magic square. But this isn’t quite correct. It is true though that two numbers that add to 20 give the same magic square. However this still gives an infinite number.
- Ask the students to write up what they have discovered.
- As the first part of the Extension problem is not so different from the original problem, most of the class might be asked to try it. Only those that can work with algebraic expressions might like to try the second part of the Extension.
Extension to the problem
Can you make up a magic square that has negative and positive numbers in it?
Can you show that the sum of a magic square is three times the centre entry?
You might expect the students at this Level to begin to work systematically. After putting in the main diagonal of –2, 4 and 10, they should experiment with different numbers in the centre top row position. We have tried 6, 8, -6 and –8 below. Remember that the sum is always 12. (Why?)
Now if we try a little algebra here we can see that there are an infinite number of these magic squares. Suppose that that centre top entry is h. Then the other entries are forced. If h is replaced by 20-h, then we get the same square. So every positive integer value over 20 gives a new magic square. There are therefore an infinite number of them.
Solution to the Extension
But why is the sum of a magic square three times the centre entry? Look at the magic square below, where all the entries are replaced by letters
Think about all of the rows, columns and diagonals through the centre square. So, if s is the sum, then
a + e + i = s
b + e + h = s
c + e + g = s
d + e + f = s
Add all of these up and you’ll get
(a + d + g) + (b + e + h) + (c + f + i) + 3e = 4s.
Now each term in the brackets equals s as they are the column sums. So
s + s + s + 3e = 4s.
This means that 3e = s just as we have told you all along!