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

Make one big triangle so the numbers that touch on the small triangles add to 10.

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 a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

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

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

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

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

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

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?

Twenty four games for the run-up to Christmas.

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

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?

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

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?

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?

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?

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?

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

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

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

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....?

How many right angles can you make using two sticks?

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?

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

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?

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

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

Place the numbers 1 to 10 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.

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

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?

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?

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

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