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

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

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?

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

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?

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

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

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

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

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 challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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?

These pictures show squares split into halves. Can you find other ways?

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

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

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

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

Can you split each of the shapes below in half so that the two parts are exactly the same?

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

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

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

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?

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

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

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

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

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

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

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.

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?

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?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

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?

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

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?

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?

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

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?

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

Can you make the birds from the egg tangram?

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?

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

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

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

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?

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.