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

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

Here are some ideas to try in the classroom for using counters to investigate number patterns.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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

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

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?

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

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.

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

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

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

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

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

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?

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

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

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?

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

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

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?

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

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

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

Can you make the birds from the egg tangram?

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

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?

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.

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 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?

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.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

How many models can you find which obey these rules?

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

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?

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?