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 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?

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

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

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

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?

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?

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?

Can you make the birds from the egg tangram?

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

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

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

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.

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?

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

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

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?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outlines of the candle and sundial?

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

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

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

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

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

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

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outlines of these clocks?

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

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

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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

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

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?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

Can you each work out the number on your card? What do you notice? How could you sort the cards?

The challenge for you is to make a string of six (or more!) graded cubes.

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?