Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

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

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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 split each of the shapes below in half so that the two parts are exactly the same?

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

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

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 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?

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

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?

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?

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?

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

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

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

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

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

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

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

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

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?

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

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

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

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.

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?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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?

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Watch this animation. What do you see? Can you explain why this happens?

Can you fit the tangram pieces into the outline of this sports car?

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