Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

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?

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?

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

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!

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. . . .

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

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?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

Exchange the positions of the two sets of counters in the least possible number of moves

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.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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.

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Use the interactivity or play this dice game yourself. How could you make it fair?

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

An interactive activity for one to experiment with a tricky tessellation

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

Train game for an adult and child. Who will be the first to make the train?

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

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

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.

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

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

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

Board Block game for two. Can you stop your partner from being able to make a shape on the board?

Can you work out what is wrong with the cogs on a UK 2 pound coin?

Can you fit the tangram pieces into the outlines of these people?

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

Can you create a story that would describe the movement of the man shown on these graphs? Use the interactivity to try out our ideas.

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

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outlines of the chairs?

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?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?