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

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?

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?

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?

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

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

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....?

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

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

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

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 1 to 8 into the circles so that the four calculations are correct?

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

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

Sort the houses in my street into different groups. Can you do it in any other ways?

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?

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 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?

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?

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!

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

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

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

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

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

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

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

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

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?

Investigate the different sounds you can make by putting the owls and donkeys on the wheel.

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

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

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

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

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?

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?

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

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

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.

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 telephone?

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

An interactive activity for one to experiment with a tricky tessellation

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

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?

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.