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

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

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?

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

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?

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?

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

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!

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

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?

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

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

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

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

Incey Wincey Spider game for an adult and child. Will Incey get to the top of the drainpipe?

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

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

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

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

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?

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?

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?

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?

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

Use the interactivity or play this dice game yourself. How could you make it fair?

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?

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

Complete the squares - but be warned some are trickier than they look!

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

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

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

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?

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

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?

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

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

An interactive activity for one to experiment with a tricky tessellation

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

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