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.

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

How many models can you find which obey these rules?

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?

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.

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

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

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?

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?

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?

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?

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?

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?

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

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

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?

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?

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?

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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

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

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?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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?

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.

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

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

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

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

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

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

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

Can you make the birds from the egg tangram?

Here is a version of the game 'Happy Families' for you to make and play.

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

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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.

We have a box of cubes, triangular prisms, cones, cuboids, cylinders and tetrahedrons. Which of the buildings would fall down if we tried to make them?