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?

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 this shape. How would you describe it?

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

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

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

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

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

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

Can you cut up a square in the way shown and make the pieces into a triangle?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

Can you fit the tangram pieces into the outline of the rocket?

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?

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

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

Can you fit the tangram pieces into the outline of Mai Ling?

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

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 these rabbits?

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?

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

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 plaque design?

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

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

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 outlines of the candle and sundial?

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

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

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

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

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

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

What is the greatest number of squares you can make by overlapping three squares?

How many models can you find which obey these rules?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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 brazier for roasting chestnuts?

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 Granma T?

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

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

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 this junk?

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?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?