Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Try out the lottery that is played in a far-away land. What is the chance of winning?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

What happens when you try and fit the triomino pieces into these two grids?

Can you find all the different ways of lining up these Cuisenaire rods?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

How many trains can you make which are the same length as Matt's, using rods that are identical?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

How many different triangles can you make on a circular pegboard that has nine pegs?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

How many different rhythms can you make by putting two drums on the wheel?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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.

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

Can you find all the different triangles on these peg boards, and find their angles?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

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

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

Find out what a "fault-free" rectangle is and try to make some of your own.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

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

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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 activity for one to experiment with a tricky tessellation

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

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

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. . . .