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.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

Make a mobius band and investigate its properties.

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

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Surprise your friends with this magic square trick.

Can you visualise what shape this piece of paper will make when it is folded?

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

Make some celtic knot patterns using tiling techniques

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

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

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

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

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

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

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.

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?

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.

Can you make the birds from the egg tangram?

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

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

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

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

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

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?

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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?

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.

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

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?

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.