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?

Watch this animation. What do you see? Can you explain why this happens?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

Can you find a way of counting the spheres in these arrangements?

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

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

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

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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Delight your friends with this cunning trick! Can you explain how it works?

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

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

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?

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

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

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

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?

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?

Find out what a "fault-free" rectangle is and try to make some of your own.

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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 you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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.

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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?

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

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

What happens when you round these three-digit numbers to the nearest 100?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

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?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these numbers to the nearest whole number?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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.

Can you describe this route to infinity? Where will the arrows take you next?

This challenge asks you to imagine a snake coiling on itself.

Try out this number trick. What happens with different starting numbers? What do you notice?

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