You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

Can you cut up a square in the way shown and make the pieces into a triangle?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

Can you make a 3x3 cube with these shapes made from small cubes?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Can you find ways of joining cubes together so that 28 faces are visible?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

This article for teachers describes a project which explores thepower of storytelling to convey concepts and ideas to children.

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

Exploring and predicting folding, cutting and punching holes and making spirals.

Can you work out what is wrong with the cogs on a UK 2 pound coin?

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

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

Here's a simple way to make a Tangram without any measuring or ruling lines.

What is the greatest number of squares you can make by overlapping three squares?

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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

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

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

Can you fit the tangram pieces into the outline of Granma T?

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

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

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

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

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

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

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

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?

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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

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