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?

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

Play this well-known game against the computer where each player is equally likely to choose scissors, paper or rock. Why not try the variations too?

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!

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?

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

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?

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?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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

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

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?

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

Exchange the positions of the two sets of counters in the least possible number of moves

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

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

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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 find out how many quarter turns the man must rotate through to look like each of the pictures.

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

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?

Train game for an adult and child. Who will be the first to make the train?

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

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

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

An interactive activity for one to experiment with a tricky tessellation

Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?

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?

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

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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?

Work out the fractions to match the cards with the same amount of money.

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!

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

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

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

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

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