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.

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

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

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?

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?

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?

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?

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

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

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?

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

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

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

How many models can you find which obey these rules?

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?

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?

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?

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

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

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

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?

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?

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

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

Can you make the birds from the egg tangram?

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

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

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

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?

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

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

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

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

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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?

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

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

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

Can you each work out the number on your card? What do you notice? How could you sort the cards?

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?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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.