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?

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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?

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?

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

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

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.

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

Make a mobius band and investigate its properties.

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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.

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

Follow these instructions to make a five-pointed snowflake from a square of paper.

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?

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

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?

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?

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

How many models can you find which obey these rules?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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

Can you make the birds from the egg tangram?

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

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

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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

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

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?

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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 smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?