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

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

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.

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

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

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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?

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

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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.

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

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

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

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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

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

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

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

Turn through bigger angles and draw stars with Logo.

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

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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?

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.

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

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.

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

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

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

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

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

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

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

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

Make some celtic knot patterns using tiling techniques

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

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?

A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

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