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

How can you make a curve from straight strips of paper?

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

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Follow these instructions to make a three-piece and/or seven-piece tangram.

Make a mobius band and investigate its properties.

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

This activity investigates how you might make squares and pentominoes from Polydron.

Surprise your friends with this magic square trick.

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Using these kite and dart templates, you could try to recreate part of Penrose's famous tessellation or design one yourself.

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

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

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

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

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

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

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

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

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?

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

Can you make the birds from the egg tangram?

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of this telephone?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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?

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

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.

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.

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

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?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

An activity making various patterns with 2 x 1 rectangular tiles.

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

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.