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?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Make some celtic knot patterns using tiling techniques

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

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

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

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

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?

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

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?

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

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

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

Turn through bigger angles and draw stars with Logo.

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

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

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

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

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?

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

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?

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 Little Ming playing the board game?

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

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

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

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.

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?

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

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

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?

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

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

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?

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

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?

How many models can you find which obey these rules?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

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?