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. . . .

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

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

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

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

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

Turn through bigger angles and draw stars with Logo.

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

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

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

What happens when a procedure calls itself?

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

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

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

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

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

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

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

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

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.

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.

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

Learn about Pen Up and Pen Down in Logo

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?

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?

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?

Make some celtic knot patterns using tiling techniques

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.

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

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

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?

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?

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

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?

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

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

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

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

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

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

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?

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

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.

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

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

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

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?