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?

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

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

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?

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?

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?

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?

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 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?

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

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

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?

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

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.

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

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

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

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

Explore the triangles that can be made with seven sticks of the same length.

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

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

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

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?

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?

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

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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

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 Little Fung at the table?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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 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?

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 outline of Wai Ping, Wah Ming and Chi Wing?

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?

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.