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.

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 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?

Which of the following cubes can be made from these nets?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

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

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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

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

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

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

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

Use the tangram pieces to make our pictures, or to design some of your own!

A game to make and play based on the number line.

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Here is a chance to create some Celtic knots and explore the mathematics behind them.

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. . . .

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

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

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?

Learn about Pen Up and Pen Down in Logo

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.

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?

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

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

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

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

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

A jigsaw where pieces only go together if the fractions are equivalent.

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

You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects.

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

What happens when a procedure calls itself?

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

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

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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

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?

Make some celtic knot patterns using tiling techniques

Galileo, a famous inventor who lived about 400 years ago, came up with an idea similar to this for making a time measuring instrument. Can you turn your pendulum into an accurate minute timer?

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

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

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.

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.