Factors and Multiples game for an adult and child. How can you make sure you win this game?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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 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?

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

How can you make an angle of 60 degrees by folding a sheet of paper twice?

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?

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

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

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?

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

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 many models can you find which obey these rules?

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?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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?

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.

A game in which players take it in turns to choose a number. Can you block your opponent?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Delight your friends with this cunning trick! Can you explain how it works?

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 birds from the egg tangram?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Reasoning about the number of matches needed to build squares that share their sides.

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

Here's a simple way to make a Tangram without any measuring or ruling lines.

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

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?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Can you fit the tangram pieces into the outline of Granma T?

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.

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

Can you fit the tangram pieces into the outline of this junk?

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?