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

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?

Can you picture where this letter "F" will be on the grid if you flip it in these different ways?

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

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

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

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

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 ...

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

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

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?

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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?

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

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.

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

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

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?

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

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?

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

Make a cube out of straws and have a go at this practical challenge.

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Reasoning about the number of matches needed to build squares that share their sides.

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

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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.

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

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

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

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

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

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

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

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

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?

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?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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?

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

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.