Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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

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

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?

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

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

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?

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

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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?

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

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

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?

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

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

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?

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

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

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

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

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 outline of the rocket?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

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

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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 this telephone?

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 ...

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

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 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 outlines of the candle and sundial?

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

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

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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

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

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 fit the tangram pieces into the outline of Granma T?

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 this shape. How would you describe it?

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