I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

Make a flower design using the same shape made out of different sizes of paper.

Can you visualise what shape this piece of paper will make when it is folded?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Can you fit the tangram pieces into the outline of this junk?

A game to make and play based on the number line.

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you fit the tangram pieces into the outline of Mai Ling?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Use the tangram pieces to make our pictures, or to design some of your own!

Can you fit the tangram pieces into the outline of this telephone?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of this plaque design?

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

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

Reasoning about the number of matches needed to build squares that share their sides.

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?