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

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

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?

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

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.

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?

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?

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

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

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?

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?

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?

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

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

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

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?

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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!

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?

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

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

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?

If you have only four weights, where could you place them in order to balance this equaliser?

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 use the numbers on the dice to reach your end of the number line before your partner beats you?

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.

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?

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?