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

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?

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

Make some celtic knot patterns using tiling techniques

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

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

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

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?

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

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

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

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

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.

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

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.

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.

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

Write a Logo program, putting in variables, and see the effect when you change the variables.

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

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?

What happens when a procedure calls itself?

Learn about Pen Up and Pen Down in Logo

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

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

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

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

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

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

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

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

Turn through bigger angles and draw stars with Logo.

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

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?

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?

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?

A description of how to make the five Platonic solids out of paper.

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

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?

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