One Wasn't Square
2 out of 3
Can you work out which three numbers were pinned onto these children's backs?
Problem
One Wasn't Square printable sheet
Mrs Morgan, the class teacher, pinned numbers onto the backs of three children: Mona, Bob and Jamie.
"Now," she said, "Those three numbers add to a special kind of number. What is it?"
Michael put his hand up.
"It's a square number," he answered.
"Correct," smiled Mrs Morgan.
Mona leaned back to look at Bob and Jamie's numbers. "Oh!" she exclaimed. "The two numbers I can see also add to a square!"
"And me!" called out Bob. "The two numbers I can see add to a square too!"
"Oh dear," said Jamie disappointedly, "the two numbers I can see don't add to a square! It's either 5 too little or 6 too big!"
What numbers did the three children have on their backs?
Getting Started
What can you deduce from what Jamie says?
Making a list of square numbers might help.
You could try finding pairs of numbers that make squares.
All the numbers, including the squares, are less than $40$.
Student Solutions
Several people sent in the correct answer to this problem. Phoebe and Lea, both at Cottesmore School tell us how they worked through it:
We found this out by adding 6 and 5 together which makes 11.
We wrote out squared numbers to 100 and then we realised that the only squared numbers with a difference of 11 between them were 25 and 36. Therefore the numbers had to add up to 36.
We then did 20 + 5 and then 11 which all makes up to 36.
Lucy and Melissa who are at Woodfall Junior School explain which number is on each child's back:
Bob's number is 11, Mona's 20 and Jamie's 5.
Mona saw 11 + 5 = 16
Bob saw 20 + 5 = 25.
Jamie looked at Bob and Mona and got 31 which is 5 less than the square number 36 and 6 more than 25.
Kevin also sent in his work on this question:
We know that Mona's number plus Bob's number is 5 less than a square and 6 more than a square. So these squares must be 11 apart. I wrote out the first few squares and saw that they get further and further apart, and the only ones that are 11 apart are 25 and 36. So Mona's number plus Bob's number is 31. We know that when you add them all up you get a square, so Jamie's number plus 31 is a square. From the hint, all the numbers are less than 40, so Jamie's number plus 31 is 36. So Jamie's number is 5. Then I found that the only way we could make the rest of the problem work is to have Mona's number as 11 and Bob's as 20 (or the other way round).
Thanks for these solutions to what can be an interesting exploration.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem invites learners to explore square numbers and the relationship between them. It can be used to talk about a trial and improvement approach and also acts as an informal introduction to algebra.
Possible approach
You could introduce this problem by having three children come to the front of the class and pinning numbers on their backs, just so that the class becomes familiar with the context. For example, you could use the numbers 2, 3 and 6. Ask each child to say something about the two numbers they can see, and invite the rest of the group to contribute statements about any of the numbers.
Introduce the problem itself - you could make a note of the key pieces of information on the board for children to refer to. Without saying much more, encourage learners to work in pairs to try and solve the problem. Indicate that you will be interested in how they reach their solution so they need to be prepared to explain this in the plenary.
Bring the class together to talk about ways of approaching this task. You can highlight particularly efficient methods, for example those which use trial and improvement in a systematic way, or those that list all the square numbers and look for differences.
Key questions
What is the gap between the two squares Jamie is talking about?
How can this help you work out the total of the two numbers Jamie can see?
What have you tried so far?
What numbers could each of the children be looking at? Why don't you try some out?
Possible extension
Children could explore the gaps between square numbers up to 100. What patterns can they discover?
Possible support
It might help for children to make a list of the square numbers, using a multiplication square if necessary. Finding some pairs of numbers that add to squares will get learners started on the problem.