Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?

What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

Play this well-known game against the computer where each player is equally likely to choose scissors, paper or rock. Why not try the variations too?

How many right angles can you make using two sticks?

An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Work out the fractions to match the cards with the same amount of money.

Can you sort these triangles into three different families and explain how you did it?

Can you hang weights in the right place to make the equaliser balance?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

Move just three of the circles so that the triangle faces in the opposite direction.

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

Take it in turns to make a triangle on the pegboard. Can you block your opponent?

A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

A game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking.

An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .

Complete the squares - but be warned some are trickier than they look!

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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

This activity challenges you to make collections of shapes. Can you give your collection a name?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Use the number weights to find different ways of balancing the equaliser.

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Twenty four games for the run-up to Christmas.

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Stop the Clock game for an adult and child. How can you make sure you always win this game?

Incey Wincey Spider game for an adult and child. Will Incey get to the top of the drainpipe?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins.

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

An interactive activity for one to experiment with a tricky tessellation

What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!