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

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 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?

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

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

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

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

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

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

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

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

Twenty four games for the run-up to Christmas.

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

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

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?

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

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

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

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

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

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

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?

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

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?

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?

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?

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

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?

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?

Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?

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?

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 complete this jigsaw of the multiplication square?

How many right angles can you make using two sticks?

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

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

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

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

Use the interactivity to sort these numbers into sets. Can you give each set a name?

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

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

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

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

Use the number weights to find different ways of balancing the equaliser.

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?

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