How many models can you find which obey these rules?

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.

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.

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?

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?

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?

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

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.

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

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?

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?

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?

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

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?

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

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

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?

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

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

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

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

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

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

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

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

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Follow these instructions to make a five-pointed snowflake from a square of paper.

Make a mobius band and investigate its properties.

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 work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Surprise your friends with this magic square trick.

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

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

Make a flower design using the same shape made out of different sizes of paper.

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

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

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

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

Follow these instructions to make a three-piece and/or seven-piece tangram.

Did you know mazes tell stories? Find out more about mazes and make one of your own.

Using these kite and dart templates, you could try to recreate part of Penrose's famous tessellation or design one yourself.

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

Here are some ideas to try in the classroom for using counters to investigate number patterns.