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

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?

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?

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

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

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

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

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

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

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?

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

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?

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

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!

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?

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

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

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

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

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?

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

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

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

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

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

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

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

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

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?

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

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?

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

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?

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

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

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?

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

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

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

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

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!

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.