Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

Are these statements always true, sometimes true or never true?

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

This problem looks at how one example of your choice can show something about the general structure of multiplication.

This investigates one particular property of number by looking closely at an example of adding two odd numbers together.

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

I am less than 25. My ones digit is twice my tens digit. My digits add up to an even number.

How many legs do each of these creatures have? How many pairs is that?

How would you create the largest possible two-digit even number from the digit I've given you and one of your choice?

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

This article for teachers describes how number arrays can be a useful representation for many number concepts.

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?

Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?

Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.

Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?

Can you find examples of magic crosses? Can you find all the possibilities?

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Can you find the chosen number from the grid using the clues?

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?