What shape and size of drinks mat is best for flipping and catching?

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

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

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

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

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

Learn about Pen Up and Pen Down in Logo

More Logo for beginners. Now learn more about the REPEAT command.

Can you lay out the pictures of the drinks in the way described by the clue cards?

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

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.

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

Make some celtic knot patterns using tiling techniques

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

This article for students gives some instructions about how to make some different braids.

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

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

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

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?

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

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

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

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

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?

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

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

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

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

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

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.

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 birds from the egg tangram?

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 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?

A description of how to make the five Platonic solids out of paper.

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?

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?

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?

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

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?

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

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

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

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

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