These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

This challenge is about finding the difference between numbers which have the same tens digit.

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?

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?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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.

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Here are two kinds of spirals for you to explore. What do you notice?

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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?

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

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 round these three-digit numbers to the nearest 100?

Stop the Clock game for an adult and child. How can you make sure you always win this game?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

An investigation that gives you the opportunity to make and justify predictions.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?