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?

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

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

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?

Can you fit the tangram pieces into the outline of these convex shapes?

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

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

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

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

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?

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

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 brazier for roasting chestnuts?

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

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

Can you fit the tangram pieces into the outline of this plaque 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.

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

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

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?

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.

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

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

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

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

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

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

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 sports car?

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

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

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 fit the tangram pieces into the outline of Mai Ling?

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

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

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

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

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

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.

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

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?

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?

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

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

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?

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

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