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!

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

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?

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?

A game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking.

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

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

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

An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .

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

Can you complete this jigsaw of the multiplication square?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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

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?

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

An interactive activity for one to experiment with a tricky tessellation

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?

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

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?

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

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?

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

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

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?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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

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.

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

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

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

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

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!

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

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....?

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

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?

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?

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

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins.

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

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

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?

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