# Number Differences

## Problem

*Number Differences printable sheet*

You might like to try A Ring of Numbers and More Rings of Numbers before this problem.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. (You must use each of the numbers once.)

Can you find some other ways to do this?

Can you put the numbers in the squares so that the difference between joined squares is even? Explain your answer.

What general statements can you make about odd and even numbers?

##### This problem is based on an idea taken from "Apex Maths Pupils' Book 2" by Ann Montague-Smith and Paul Harrison, published in 2003 by Cambridge University Press.

## Getting Started

Try putting any number in one square to start with.

What numbers could go in the squares which join this first square? What do all these options have in common? Can this help you find other arrangements of the numbers?

You might like to print out this sheet of blank grids if you are not using the interactivity.

## Student Solutions

Thank you to everybody who sent us their ideas about this problem. The Maths Club from St Peters Catholic Primary school sent in this explanation:

It is easy to make all the differences odd - by using consecutive numbers forwards and backwards. We need to have at least one odd number in each row or column. We could not find a solution where all the differences were even, we think you would need one more odd or even number to fullfill that criteria, or if we could use one of the numbers twice.

Good ideas! I wonder which rows and columns need to have one odd number in them, and which need to have more than one? I'm also not sure that just adding in one more number will help us make all of the differences even - I think we might need to add in some more numbers to make that possible.

Springvale Primary School sent in solutions from: Nathan, Meg, Hannah, Martha, Oliver, Anton, Mayra, Alice, Cameron, Laurence, Georgina, Eddie, George, Freya, Emily, Abigail, Ewan, Lara, Jack, Emilia, Scarlett, Cara and James.

Alice and Cameron's picture was:

You've found a lot of solutions! I wonder if we could work systematically to find all of the possible solutions? Or do you think there would be too many possibilities?

Freya's picture was:

Well done for finding a general solution, Freya. The children in Year 5 at Heathlands Primary Academy also described the same general solution:

We found out that we needed to place the numbers so that even numbers were connected to odd numbers in order to have an odd difference. This meant we could mix the numbers up and find multiple ways of arranging the numbers in the grid.

E.g.:

5 4 7

6 9 2

1 8 3

We then tried the extension task of trying to arrange the numbers so that the difference between the connected numbers was always even. This was harder because we found that we either had to match even with even or odd with odd. We didn't have enough even or odd numbers to make the grid. We couldn't connect even and odd numbers as that would always leave an
odd difference. We could find ways to complete this using our own numbers between 0-20 - either by using all odd numbers or all even numbers.

Ayati from Wimbledon Chase Primary School in England had the same idea about making all of the differences even:

I moved on to the second question. This time there should have been an even difference. I tried many different possibilities but finally had to conclude that this was impossible. I believe the reason for this is because odd and odd make even and even and even make even, however odd and even make odd so the whole shape would either have to be odd or even. There are not enough numbers to do this so it is impossible to do with the numbers 1-9.

That's right, Ayati - we would either need all of the numbers to be even or we would need all of the numbers to be odd.

## Teachers' Resources

Why do this problem?

This problem encourages pupils to form early stages of proof by using their knowledge of odd and even numbers to construct mathematical arguments. It is a good context for generalisations.

Possible approach

Ideally, this activity should follow on from Ring A Ring of Numbers and More Numbers in the Ring so that minimal introduction will be needed. However, if learners have not worked on the previous two tasks, you could show
them the grid on the interactive whiteboard and write in numbers (in whatever way you like) and ask them to talk to a neighbour about what they see. Sharing children's observations is likely to bring up odd and even numbers so that you can offer the challenge as it is written.

Give pairs time to work on the activity and draw the whole group together when appropriate to share any insights. It may be useful for children to have a sheet with blank grids in order to try out their ideas. Digit cards would also be handy.

You might like to invite children to find more than one way of arranging the digits for odd differences. When it comes to the challenge of arranging the digits so that the difference between joined numbers is even, look out for those children who have a hunch it is impossible. Encourage them to articulate why they think it cannot be solved. You may find that a whole-group
discussion is a fruitful way of coming to a good explanation.

Key questions

What have you tried so far?

Are there any other ways to do it?

How do you know this is impossible? Perhaps we just haven't found the one way that works?

Possible extension

You can assess children's understanding of the situation by giving them a new set of numbers to arrange. For example, what would happen with 2, 3, 4, 5, 6, 7, 8, 9 and 10? Are there any sets of numbers that would make both parts of the problem possible? This will encourage them to generalise the situation beyond numbers 1 to 9.

Alternatively, you may like to probe children's understanding of the way odd and even numbers behave. How do they know that two odd numbers will always have an even difference? Can they convince you that it will always be the case? They may well be able to create pictures that form a proof.

### Possible support

Many children will find it beneficial to work on Ring A Ring of Numbers and More Numbers in the Ring before tackling this problem.