Little magic squares

Student Activity

Tui has just discovered magic squares.

She decides to make all of the magic squares that she can just using the numbers 1, 2 and 3.

How many can she make?

It takes her quite a while because she doesn't know that the sum of a magic square is always three times the number in the centre.

Specific Learning Outcomes
Recall addition facts
Know the idea of, and be able to construct, magic squares.
Description of Mathematics

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. It is likely that students have already encountered magic squares. The problems in this sequence give students the opportunity to use known numerical or algebraic concepts.

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.
This is proven 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 second in a sequence of problems on magic squares. The first of these is A Square of Circles (and is also at Level 2), and no attempt is made to explore magic square properties here.

Magic square problems at Level 3  use 3-digit numbers (Big Magic Squares) and decimals (Decimal Magic Squares). At Level 4, Negative Magic Squares, uses negative numbers and Fractional Magic Squares uses fractions. The Magic Square, Level 5 shows why three times the centre number is equal to the sum of the magic square. Finally, Difference Magic Squares at Level 6, looks at an interesting variation of the magic square concept.

Activity

The Problem

Tui has just discovered magic squares. She decides to make all of the magic squares that she can just using the numbers 1, 2 and 3. How many can she make?

It takes her quite a while because she doesn't know that the sum of a magic square is always three times the number in the centre.

Teaching sequence

  1. 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.
  2. Show students a magic square such as the one below. Ask them to think about why it is called a Magic Square.
    6
    1
    5
    3
    4
    5
    3
    7
    2
  3. Have them 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.
  4. Have students tell you why it is called a Magic Square. (The sum of a magic square is the common sum of the rows, columns and diagonals.)
  5. Read Tui’s problem together. Have them check on the magic square above to confirm that 3 x the centre number (4) really does equal the sum (12). 
  6. Ask them to go away in pairs and see how many magic squares they can find using only the numbers 1, 2 and 3.
  7. Get some of the pairs to report back. Can they prove that the arrays they have produced are magic squares?
  8. Have students work on the Extension problem as appropriate.

Extension

How many magic squares would Tui have made if she had only the numbers 7, 8 and 9 to use?

 

Note: This problem can be done with any three consecutive numbers. You will only get the four answers.

If you use three consecutive even numbers or three consecutive odd numbers, again you only get four magic squares.

You might like the students to try other three numbers to see how many magic squares that they can find. You could find more or you may find less.

Solution

Students are unlikely to solve the problem the way shown below.  Students are more likely to use guess and check and to stumble across the final set of solutions. However, a systematic approach is shown here to show that Tui should have found only 4 magic squares.

You may however want to have your students see that there is a systematic way of getting the four answers.

Being systematic in this problem can mean choosing different numbers for the centre square. So the centre square could be 1, 2 or 3.

centre square = 1: This means that the sum of the magic square has to be 3. This sum can only be made by using three 1s. So this magic square consists of all 1s. Call this square A.

1
1
1
1
1
1
1
1
1

centre square = 2: This means that the sum of the magic square has to be 6. Now 6 can only be made with 1 + 2 + 3 or 2 + 2 + 2. One way to get a magic square here is for all of the entries to be 2. This is not very interesting but it does give another magic square that we will call B.

2
2
2
2
2
2
2
2
2

Now suppose that the centre square (2) is used with 1 and 3 somewhere to get the sum of 6. Because of the symmetry of the square, we can assume without loss of generality that this is either done on the main diagonal or on the vertical column through the centre.

In the first case, the middle square in the top row is either a 2 or a 3. (It can’t be 1 because then the row sum would not be 6.) We follow through these two situations.

In the ‘2‘ case, we have to have a 3 in the top right-hand square. But then the last column can’t sum to 6.

12 
 2 
  3
123
 2 
  3
123
 2?
  3

In the ‘3’ case, the 2 in the top row and the 1 in the middle column are forced. This then means that there has to be a 1 in the middle square of the last column. This forces the 3 and 2 in the first column. A quick check shows that we have another magic square. Call this magic square C.

13 
 2 
  3
132
 2 
 13
132
 21
 13
132
321
213

Now we have to worry about the 1, 2, 3 being in the centre row. Because of the symmetry of the square, we can assume that there is a 2 in the top left-hand square and a 3 in the top right-hand square. This forces the two 1s as shown and then the final 3 falls into place. A final check shows that this is a magic square.

 1 
 2 
 3 
213
 2 
 3 
213
 2 
 32
213
 21
132
213
321
132

If we rotate this last magic square through 90° , then it looks exactly the same as C. So we don’t get a new magic square this way.

centre square = 3: This means that the sum of the magic square has to be 9. This can only be done if the three numbers that make up a row or a column are all 3s. So we get another uninteresting magic square that we will call D.

3
3
3
3
3
3
3
3
3

Tui should have found four magic squares. These are the ones we have called A, B, C and D.

Solution to the Extension

Tui would have found only four magic squares this time too. Using exactly the same method as before she would have come up with the following answers.

777
777
777
888
888
888
999
999
999
879
987
798
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