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.

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

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

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

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 Little Fung at the table?

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

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

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

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

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

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

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

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

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

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

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 outlines of the workmen?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of this goat and giraffe?

Can you fit the tangram pieces into the outline of this plaque design?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

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

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

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

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?

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

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?

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

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?

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

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

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

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

Can you make the birds from the egg tangram?

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 logically construct these silhouettes using the tangram pieces?

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

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

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?

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

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