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?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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?

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?

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 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?

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?

These pictures show squares split into halves. Can you find other ways?

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?

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How many models can you find which obey these rules?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

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

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

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

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

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

Can you make the birds from the egg tangram?

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

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

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 this goat and giraffe?

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 this plaque design?

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 describe a piece of paper clearly enough for your partner to know which piece it is?

Can you split each of the shapes below in half so that the two parts are exactly the same?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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

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

Can you make five differently sized squares from the tangram pieces?

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

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 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 this telephone?

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 the child walking home from school?

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