Magic Squares

Task 92 ... Years 4 - 10

Summary

A classic problem which is presented differently here because there are no clues. Sure, all the rows, columns and the two leading diagonals must add to the same number, but the first problem may by to calculate that number. Once solved the card invites the students to explore several What happens if..? questions.
 

Materials

  • Tiles numbered from 1 - 16
  • 3 x 3 and 4 x 4 playing boards
  • Marker and cloth

Content

  • basic number skills
  • problem solving strategies, especially breaking the problem into parts and working backwards.
Magic Squares

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

It's likely students will begin this problem using a guess and check strategy. Then they might begin to see that there are 8 lines which have to have the same total, so they need 8 number triples (made using the digits 1 to 9) that sum to the same number. What might that number be?

Working backwards can help to find it out. Assume the problem has been solved. Then the three columns will each add to the same total. If those totals were then added that would be the same result as adding the nine separate digits because each digit is counted once in the three separate additions. The sum of 1 to 9 can be found in various ways. It's forty-five. So the sum of the three column totals is 45 and column totals are equal, so each one must be 15.

The next step in the problem might be to list all the triples that add to 15:

(9,5,1)  (9,4,2)  (8,6,1)  (8,5,2)  (8,4,3)  (7,6,2)  (7,5,3)  (6,5,4)
There are 8 of them, which is convenient because 8 lines of the magic square have to be equal. Four of these lines pass through the same square, the centre, which means one number must be in four of the triples. Only 5 satisfies that condition, so 5 must go in the middle and we now know the other parts of each triple it is lined up with horizontally, vertically and on the two diagonals. We don't know which 5 triple goes in which of these lines yet, but by trying out possibilities and considering the triples that must be consequences in the border lines, we can find out.

Eventually a solution will be found and it may appear in one of several variations:

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

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

Studying this set of solutions more closely reveals that those in the bottom line are reflections of those above. Further, looking along the top row reveals that each solution is the one to its left rotated 90. So really there is only one solution and the others can be created from it.

Another strategy for tackling the problem is to break it into smaller parts. For example:

  • Firstly, set the tiles into three separate columns that each total 15. [Don't look at anything else - just columns.]
  • Now swap numbers within columns. This will not alter the column total, but changes the rows. Keep shuffling until the rows also add to 15.
  • Now focus on diagonals. By swapping two rows or swapping two columns, you will not alter row or column totals but you will change the diagonals. Keep swapping until the diagonals also work.
This strategy probably is easier to apply to the 4 x 4 case than the strategy of finding and managing all the quadruples that add to its magic total.

The remainder of the task allows students to discover how to make new magic squares from old ones. It is based on the idea that if the same operation as is performed on every number in the square the result should also be magic. So, if every number was divided by 3, say, we would have a Fraction Magic Square, which is Task 37.

Extension
What happens if the condition is to arrange the digits 1 - 9 into the 3 x 3 square so that no row, column or leading diagonal has the same total. That is, each of the 8 totals is different.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

Ask the students to tear a piece of paper into nine parts and number them. Now set the problem. It helps if you have nine large tiles yourself, about 20cm2, to demonstrate the problem on the community floorboard. Maths300 provides a lesson plan, supported by software, that details how this lesson sequence can be carried through.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 98, Magic Squares, which includes an Investigation Guide and companion software.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Magic Squares task is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 9 & 10

The Magic Squares lesson is an integral part of::

  • MWA Number & Computation Years 9 & 10

Green Line
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