Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

Build a scaffold out of drinking-straws to support a cup of water

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

What shape would fit your pens and pencils best? How can you make it?

What shape and size of drinks mat is best for flipping and catching?

Make some celtic knot patterns using tiling techniques

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Can Jo make a gym bag for her trainers from the piece of fabric she has?

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

Make an equilateral triangle by folding paper and use it to make patterns of your own.

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

The challenge for you is to make a string of six (or more!) graded cubes.

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

Make a clinometer and use it to help you estimate the heights of tall objects.

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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

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.

This article for students gives some instructions about how to make some different braids.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

Did you know mazes tell stories? Find out more about mazes and make one of your own.

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

A game in which players take it in turns to choose a number. Can you block your opponent?

Learn about Pen Up and Pen Down in Logo

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

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

Learn to write procedures and build them into Logo programs. Learn to use variables.

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

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

More Logo for beginners. Now learn more about the REPEAT command.

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

Turn through bigger angles and draw stars with Logo.

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.

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 make the birds from the egg tangram?

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

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

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?

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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?

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?