Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

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?

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

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

What is the greatest number of squares you can make by overlapping three squares?

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?

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

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

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of 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?

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?

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.

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

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

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

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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?

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

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

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?

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.

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?

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.

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

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

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?

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

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 junk?

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

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?

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

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.

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

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.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?