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?

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?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Here is a version of the game 'Happy Families' for you to make and play.

These practical challenges are all about making a 'tray' and covering it with paper.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Can you make the birds from the egg tangram?

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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 make dice stairs using the rules stated? How do you know you have all the possible stairs?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How many models can you find which obey these rules?

Delight your friends with this cunning trick! Can you explain how it works?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

Factors and Multiples game for an adult and child. How can you make sure you win this game?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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.

An activity making various patterns with 2 x 1 rectangular tiles.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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 ...

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

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

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

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

What do these two triangles have in common? How are they related?