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?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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.

What is the best way to shunt these carriages so that each train can continue its journey?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted 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?

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

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?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

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?

Which of these dice are right-handed and which are left-handed?

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

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

An activity centred around observations of dots and how we visualise number arrangement patterns.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Exchange the positions of the two sets of counters in the least possible number of moves

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

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

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

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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

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?

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

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

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 Little Ming playing the board game?

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

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?