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?

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

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

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

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

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?

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 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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

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

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

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

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

Our 2008 Advent Calendar has a 'Making Maths' activity for every 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.

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

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

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

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?

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

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

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.

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?

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

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

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

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

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

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

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....?

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.

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!

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?

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

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

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

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

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?

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

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

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