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?

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?

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

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

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

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

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

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?

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

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

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

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

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?

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

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

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?

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?

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?

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

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

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

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?

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?

Can you fit the tangram pieces into the outlines of the candle and sundial?

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

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

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

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 everybody knows. You can play with a friend or online. If you play correctly you never lose!

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!

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

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

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?

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

Can you fit the tangram pieces into the outline of the child walking home from school?

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

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

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

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

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!