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?

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

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?

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!

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

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.

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

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.

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

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?

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?

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?

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?

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

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

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.

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?

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.

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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

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

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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

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

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

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 Granma T?

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 the candle and sundial?

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

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?

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

How many different triangles can you make on a circular pegboard that has nine pegs?

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

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

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?

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

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

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

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

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

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

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

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