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

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?

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

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

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

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

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

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

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?

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

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?

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

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?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

How many models can you find which obey these rules?

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

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?

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.

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

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?

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

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?

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

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

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

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

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

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

Exploring and predicting folding, cutting and punching holes and making spirals.

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

Can you deduce the pattern that has been used to lay out these bottle tops?

What shape is made when you fold using this crease pattern? Can you make a ring design?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Can you make the birds from the egg tangram?

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

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?

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

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?

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

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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.

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