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!

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

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?

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

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

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

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?

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 which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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

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?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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?

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

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

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....?

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?

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.

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

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

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

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?

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

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

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 put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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

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

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Try out the lottery that is played in a far-away land. What is the chance of winning?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?

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

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

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

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

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?