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

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?

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....?

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 draw on the dotty grid which each have one dot in the middle?

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

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

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

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

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

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

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

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

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?

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?

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

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?

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

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

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

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

Terry and Ali are playing a game with three balls. Is it fair that Terry wins when the middle ball is red?

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

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

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Can you hang weights in the right place to make the equaliser balance?

Use the information about Sally and her brother to find out how many children there are in the Brown family.

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

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?

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

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

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?

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?

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

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?

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

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

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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

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

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