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?

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?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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.

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

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

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?

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

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 investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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?

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

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

How many models can you find which obey these rules?

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?

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

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

Can you make the birds from the egg tangram?

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

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

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

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

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?

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

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?

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

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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?

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

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

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

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?

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

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

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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.

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

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

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?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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 ...