Square Pairs

Task 216 ... Years 4 - 12

Summary

Perhaps the most exciting thing about this problem is that it is exactly as worked on by two modern mathematicians, Johnston Anderson and Andy Walker, University of Nottingham, and that they suggested it as an appropriate challenge for school students. As well as other aspects of the problem they saw that ...more importantly, (it gives) an opportunity to try to construct proofs.

Students investigate any sequence of numbers 1 to n, where n is even and terms have a common difference of 1, by finding a way to make pairs so that all the numbers in the sequence are square paired. Square pairing is the process of selecting two numbers whose sum is a square number. You can see this activity being started in the Cube Tube video: An Ocean of Possibilities.

 

Materials

  • 20 tiles numbers 1 to 20

Content

  • concept of proof
  • history of mathematics
  • numbers, odd & even
  • numbers, square
  • reasoning
  • recording mathematics
Square Pairs

Iceberg

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

The problem begins by exploring the list 1 to 16. Students generally start with a guess and check approach. Perhaps the turning point is when they realise that 16 is already a square, so what can it be paired with to make a total that is a square? Whatever it is the total must be a square number higher that 16, which means the total must be 25 and the partner must be 9.

  • Why can't the total of 16 with another number in the list reach the square number after 25, which is 36?
So one square pair from this list must be (9, 16), because 16 cannot be paired with anything else. The consequence of this pairing is that 16 and 9 can no longer be used. The list is now:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
So, it was fruitful beginning at the larger end of the sequence, so perhaps they will continue from there.

What happens if we now look at 15? Uh oh! it could pair with 1 or 10. (16 can't be used in a pair, but that doesn't mean it can't be a total.) Students now have to follow each pathway until they either create 8 pairs, or can go no further. Recording is essential and using the tiles certainly makes the job easier. The reasoning might continue like this:

If 15 pairs with 1...
(9, 16) (1, 15) 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14
...which gives two choices for 14, take the smaller
(9, 16) (1, 15) (2, 14) 3, 4, 5, 6, 7, 8, 10, 11, 12, 13
...which gives two choices for 13, take the smaller and remember to come back and check the growing number of other choices. (9, 16) (1, 15) (2, 14) (3, 13) 4, 5, 6, 7, 8, 10, 11, 12
...which leaves only 1 choice for 12
(9, 16) (1, 15) (2, 14) (3, 13) (4, 12) 5, 6, 7, 8, 10, 11
...which leaves only 1 choice for 11
(9, 16) (1, 15) (2, 14) (3, 13) (4, 12) (5, 11) 6, 7, 8, 10
...which leaves only 1 choice for 10 but unfortunately that will leave 7 & 8, which can't be square paired.

So now the students could go back and carefully explore every possible alternative pathway under the heading of 15 pairing with 1 and then repeat the process with 15 paired with 10. One of these pathways must lead to a correct pairing, or perhaps several correct pairings.

Or...

They might be helped to look a little deeper into the problem with a question like:

If I choose one of these tiles, can you tell me the only other ones that can pair with it? It might help if you make some sort of table of which numbers pair with which.
This could lead to noticing two more useful things:
  • 8 can only pair with 1.
  • Once 9 has been paired with 16, 7 can only be paired with 2.
Following this must pair line of thought leads to this unique solution, although is still the possibility of making wrong choices along the way:
(9, 16) (2, 7) (11, 14) (4, 5) (12, 13) (1, 8) (3, 6) (10, 15)
Rearranging this list as:
(1, 8) (2, 7) (3, 6) (4, 5) || (9, 16) (10, 15) (11, 14) (12, 13)
the answer to the second question on the card becomes clear.

Students now have enough background to explore other examples. Question 3 begins this investigation and the only two new lists within 1 - 20 that will square pair are 1 - 14 and 1 - 18. However, students are being prepared for the challenge of finding lists which don't work, because in searching for these two they will discover some lists that can't work. One of these is 1 - 20 itself. Having found some lists that don't work, the challenge is to rediscover all seven lists which Johnston and Walker proved cannot square pair.

More importantly, the challenge is to explain why they can't square pair. It is not the same reason for each of the seven lists.

Extension

What happens if we list from 1 to an odd number? First response might be that we can't make pairs; there will always be one tile unpaired. Then, what happens if we list from 0 to an odd number. Are there any lists that square pair?
(Actually, there are only three 0 to n lists which don't square pair?)

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.
   

This is a great opportunity to combine several teaching techniques. Begin the lesson by asking students to each tear a piece of paper into 9 pieces. Then Partner A numbers their set 1 - 9 and Partner B numbers theirs 10 - 18. For yourself prepare in advance a set of cards about 20cm x 20cm numbered from 1 to 18, or simple user a marker on a sheets of paper at the same time as the students are making their sets.

Invite 18 students to come forward to receive a card.

I have a little puzzle for you. It is possible for each of you to find a partner so that when you add your two numbers, the total is a square number. Let's see if you can work out how.
As pairs are made, ask them to separate from those who are still seeking their partner. Soon it will become clear that those remaining aren't able to make squares.
But it can be done ... so I guess some couples will have to divorce.
It may or not work out that the actors complete the puzzle. But it doesn't matter because either way the players return to their seats and partners try to solve the puzzle themselves using the numbered paper tiles. Discuss and record information students discovered which helped with the solution, then begin an exploration of other lists.

As students begin to feel that it will work for any number, introduce the Johnston & Walker discovery and shift the investigation to finding the seven that don't work and explaining the reasons why.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 140, Square Pairs which includes companion software for exploring every possible case from 1 to n, where n is less than or equal to 99.

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 Square Pairs task is an integral part of:

  • MWA Number & Computation Years 9 &10

The Square Pairs lesson is an integral part of:

  • MWA Number & Computation Years 9 &10

Green Line
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