Find the difference
Problem
Find the Difference printable sheet
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
For example:
You could use this interactivity to try out your ideas:
Can you find more than one solution?
Can you find all the possible solutions? How do you know you have found them all?
Getting Started
You could use numbered counters that can be moved about if you're not using the interactivity.
Where might the largest number go?
You might find this sheet of blank pyramids useful.
Student Solutions
Thank you to everybody who sent in their solutions to this problem. Lots of children sent us one or two solutions, but only a few children managed to find every solution. Well done to all of you!
Dhruv from The Glasgow Academy in Scotland sent in these two solutions along with some reasoning about where the 5 and 6 could go:
That reasoning is really clear, Dhruv - you've convinced me that 6 has to be on the bottom row and that 5 can't go at the top.
We received a lot of solutions from the children at St Helen's C of E School in England. Sonny, Reuben and Rio found a similar solution to one of Dhruv's, and they explained their process:
1 This is the correct solution.
4 3
6 2 5
First, we tried 1 at the top.Then we put 3 and 2 on the middle section. Then we put 6 and 4 at the bottom and knew that it was wrong.
On our 2nd try we put 1 at the top, 4 and 3 in the middle and 6, 2 and 5 at the bottom which was right!
6 cannot be put in the top or middle sections because it is the highest number.
5 can only be in the bottom or middle sections.
Ralph found a different solution and noticed something about where the 3s could go:
2
5 3
6 1 4
I knew that 6 had to be at the bottom because there was no bigger number to make a difference.
1 was the first number I tried next to 6 and it worked!
I knew 3 couldn't go next to 6 because we would need two 3s.
Amelia-May, Amélie and Joanne found another new solution. They noticed the same thing as Ralph, and also noticed something about the 4 and 2:
We found out that 6 could only be placed in the bottom row because it was the largest number and it couldn't be the difference between any other numbers.
Our 5th and 6th attempts were the only ones that were effectively successful.
Numbers that we found couldn't go together were 2 and 4 because there is a difference of 2 and there was only one 2 on top of that 3 and 6 because the difference is 3 and again there is only one of each number.
We found 2 solutions:
1
4 3
6 2 5
and
3
1 4
5 6 2
Thank you all for sharing your ideas with us, and thank you as well to Albie, Adam, Riley, Sam, Olivia, Tommie, Frances, Lucy, Rosie and Ophelia who all sent in very well-reasoned solutions. We also received some similar solutions from: Nivaan Haluvaagilu from Ekya School JP Nagar in India; all the children at Cromwell Community College Primary Phase in the UK; May, Mia, Joanna and Jolene from Falcons School for Girls; and Micah, Ruth, Blythe and Khalil from Bay Vista Fundamental in the USA.
Alba from Barton C of E School in Cambridge, UK managed to find every possible solution. Alba said:
I used my number bonds.
I used counters to help.
You can’t put 6 and 3 together because you make 6 as 3 + 3
6 can't go up high because it’s the biggest number.
I highlighted the triangles that matched and it helped me to find the missing one.
Well done, Alba! This is an excellent example of how we can work systematically to find a solution that we've missed out. Can you see how Alba has decided which triangles 'match'?
We also received a full solution from Vansh, Eshaan, Vraj, Rishaan, Samaira, Renah, Udit, Uday, Viha, Gowri, Arya, Krishna, Rivaan, Miraya and Hiren at Ganit Kreeda, Vicharvatika in India. Their observations were:
6 cannot go in the topmost two rows as 6 being the biggest number it is not possible to find 2 numbers from 1 to 6 whose difference is 6. That means 6 can occupy only boxes in the bottom row.
5 cannot come at the topmost layer as we cannot use 6 in the second row.
4 cannot come at the topmost square as the only way to get 4 is by 5-1 in the 2nd row. To get 5 in the second row, the only way is 6-1. As we cannot repeat the numbers (1), we cannot use 4 at the top.
Only smallest 3 numbers (first 3 numbers) can come at the top.
Based on our observations we made a table as follows:
Kids analysed the solution from the 3-layer pyramid with 3 at the top.
They noticed that by changing the positions of 1 and 4 with 5 and 2, we can get these 2 different solutions.
Rishaan beautifully explained the reason.
4-1 = 5-2 =... so we can change their positions.
Also, 6-1=5 & 6-5=1 and 6-4=2 & 6-2=4
We can change their relative positions.
Well explained, Rishaan, and well done to all of you for your hard work on this. To see more ideas from this group, including some algebraic reasoning from Samaira, take a look at Ganit Kreeda's full solutions.
Teachers' Resources
Find the Difference
Find the Difference printable sheet
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
For example:
You could use this interactivity to try out your ideas:
Can you find more than one solution?
Can you find all the possible solutions? How do you know you have found them all?
Why do this problem?
This problem is a challenging way of practising subtraction at the same time as being logical about arranging the numbers. The idea of 'difference' can be hard for children to grasp and this problem is an ideal way of coming to terms with it. You could also use this problem to focus on how children record their workings.
Possible approach
You could start by putting the numbers in any places in the pyramid and asking children to describe what they see. They may notice some accidental number patterns, but also which numbers are used. Then put two numbers into the pyramid, for example 4 and 5 in the bottom row, next to each other. Introduce the idea of the problem and invite pupils to suggest what number should go above the 4 and 5. You could repeat this a few times with different pairs of numbers so that children are happy with 'difference' and with the way the pyramid will be structured. You may need to emphasise that the smaller number will be taken away from the larger number each time.
Key questions
What do we need to do to find the difference between two numbers?
How will you keep track of what you have tried?
Possible extension
Possible support
Many children would benefit from using numbered counters that can be moved about if they don't have access to the interactivity. Some children may need to use manipulatives to support them in finding the differences.