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?

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

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?

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?

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!

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

An environment which simulates working with Cuisenaire rods.

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

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?

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

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

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

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

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

Here is a chance to play a version of the classic Countdown Game.

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?

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

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....?

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?

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?

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

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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?

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

Can you complete this jigsaw of the multiplication square?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

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

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

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

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

Use the number weights to find different ways of balancing the equaliser.

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

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

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

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

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

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

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

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

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

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?

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

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

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?