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?

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

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

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

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....?

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.

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?

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?

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

Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

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

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

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?

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?

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

These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

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?

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.

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

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?

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

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?

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

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

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?

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

Can you sort numbers into sets? Can you give each set a name?

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

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

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

How many different sets of numbers with at least four members can you find in the numbers in this box?

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

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

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.

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

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.

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?

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

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

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

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

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