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Level Three > Number and Algebra

Decimal Magic Squares

Keywords:

Achievement Objectives:

Achievement Objective: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.

Specific Learning Outcomes:

Use addition with decimals

Know the idea of, and be able to construct, magic squares

Description of mathematics:

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.

4	1	7
7	4	1
1	7	4

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 the Level 4 lesson Negative Magic Squares in this sequence and in the Level 5 lesson (The Magic Square).

This problem is the fourth in a 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 – this lesson and Big Magic Squares (which uses 3-digit decimal numbers).

At Level 4, Negative Magic Squares, uses negative numbers and 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.

Required Resource Materials:

Copymaster of the problem (English)

Copymaster of the problem (Māori)

Activity:

The Problem

Tui has begun to like magic squares. She decided to make all of the magic squares that she could using the numbers 2.0, 2.2, 2.4, 2.6 and 2.8. How many could she make if she used each number at least once in the square?

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.

Teaching sequence

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.

6	1	5
3	4	5
3	7	2

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?
Ask the students to write up what they have discovered.
As the Extension problem is not so different from the original problem, most of the class might be asked to try it.

Extension to the Problem

Make up a magic square using at least four decimal numbers of your own choice. None of these magic squares can have the same number in each place.

Solution

We should say right at the start that we don’t expect the students to solve the problem the way we do below. We would expect the students to use guess and check and to stumble across the final answer. We have, however, done this problem very systematically here so that you can see and be absolutely sure that Tui should have found only one magic square. (The other three that they might find use only one number and here we said that Tui had to use each number at least once.) It might be worthwhile trying to lead the class into seeing that there is a systematic way of getting the answer.

Because the sum of the magic square is three times the centre entry, we consider the possible numbers that could go in the centre.

centre square = 2.0: Here the sum has to be 6. But the only way to get a sum of 6 from the numbers provided is to use 2.0 three times. But then we would have to have a magic square with 2.0 in every place. This is against the rules here.

centre square = 2.2: This forces us to have a sum of 6.6. This can only be done using 2.0, 2.2 and 2.4. We can get an answer here by replacing 7 by 2.0, 8 by 2.2 and 9 by 2.4 in the fourth answer in the Little Magic Square. Call this magic square A.

centre square = 2.4: Here the sum has to be 2.4. No we can make 3 ´ 2.4 = 7.2 in a number of ways. These are by using 2.0, 2.4 and 2.8; 2.0, 2.6 and 2.6; 2.2, 2.2 and 2.8; 2.2, 2.4 and 2.6; and 2.4, 2.4 and 2.4. (These are the only possibilities. This can be shown by making a systematic list.)

Now we have to continue systematically.

First suppose that the main diagonal is 2.0, 2.4, 2.8. Then what can go into the centre top row square? It can’t be 2.0 (or 2.2) since there is no number that, along with 2.0 and 2.0 (2.2) gives 7.2. But it can be 2.4, 2.6 or 2.8. We take each of these in turn.

In the first two cases we get a problem where the question marks are. In the third case we get a magic square (B).

2.0	2.4	2.8
2.4	??
2.8

2.0	2.6	2.6
2.4	??
2.8

2.0	2.8	2.4
2.8	2.4	2.0
2.4	2.0	2.8

So now suppose that the main diagonal is 2.2, 2.4, 2.6. Then the top row centre square has to be 2.2, 2.4, 2.6 or 2.8. Here we seem to get three answers but the first and third of these are the same (flip the third one about the main diagonal). So we get two new magic squares here, C and D.

2.2	2.2	2.8
2.4	??
2.6

2.2	2.4	2.6
2.8	2.4	2.0
2.2	2.4	2.6

2.2	2.6	2.4
2.6	2.4	2.2
2.4	2.0	2.6

2.2	2.8	2.2
2.4	2.4	2.4
2.6	2.0	2.6

This means that the only possibility left is 2.4, 2.4, 2.4 down the main diagonal. This leaves the five numbers 2.0, 2.2, 2.4, 2.6 and 2.8 as possibilities for the top row centre square. But putting 2.4 in there, we are forced to have all of the entries equal to 2.4, so we omit this possibility.

2.4	2.0	2.8
2.8	2.4	2.0
2.0	2.8	2.4

2.4	2.2	2.6
2.6	2.4	2.2
2.2	2.6	2.4

2.4	2.6	2.2
2.2	2.4	2.6
2.6	2.2	2.4

2.4	2.8	2.0
2.0	2.4	2.8
2.8	2.0	2.4

Now at first sight it looks as if we have found four more answers. However, the first and last one both have 2.0, 2.4 and 2.8 along a diagonal and so can be rotated to give the only answer with 2.0, 2.4 and 2.8 along the main diagonal. And the second and third are the same – just flip the third one about the main diagonal. So we end up with only one new answer E (the second one).

centre square = 2.6: This forces the sum of the magic square to be 7.8. The only way to get 7.8 here is to use 2.4, 2.6 and 2.8. This leads us to the new square F, which can be obtained from the 7, 8, 9 square in Little Magic Squares by changing 7 to 2.4, 8 to 2.6 and 9 to 2.8.

centre square = 2.8: This forces a sum of 8.4 but this can only be done with all entries equal to 2.8.

So we get 6 answers, A, B, C, D, E and F.