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?

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?

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?

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?

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

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?

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?

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

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

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?

How many models can you find which obey these rules?

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?

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

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 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.

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

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

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

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

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?

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

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?

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

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?

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

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

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.

Can you make the birds from the egg tangram?

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

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

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

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

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

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?

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?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

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

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?

For this activity which explores capacity, you will need to collect some bottles and jars.

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