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?

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

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

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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

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

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

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

Can you put these shapes in order of size? Start with the smallest.

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

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?

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

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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

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

What shape is made when you fold using this crease pattern? Can you make a ring design?

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?

Make a flower design using the same shape made out of different sizes of paper.

Make a cube out of straws and have a go at this practical challenge.

Can you deduce the pattern that has been used to lay out these bottle tops?

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?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you see which tile is the odd one out in this design? Using the basic tile, can you make a repeating pattern to decorate our wall?

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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

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?

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?

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

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?

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

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?

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.

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

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

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?

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

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

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

Can you make the birds from the egg tangram?

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.

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!