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

Make some celtic knot patterns using tiling techniques

Make a mobius band and investigate its properties.

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.

Surprise your friends with this magic square trick.

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?

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

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

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

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

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

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

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

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

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

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

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?

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

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

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?

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?

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?

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

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

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?

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

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?

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.

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

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

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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

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?

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

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

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.

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

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Here's a simple way to make a Tangram without any measuring or ruling lines.

Can you make the birds from the egg tangram?

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?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?