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

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

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

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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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?

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

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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

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?

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?

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!

An environment which simulates working with Cuisenaire rods.

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

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

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

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

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

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

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

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 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?

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

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

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

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?

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?

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

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

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

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

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

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

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

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

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

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

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

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?

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?

Find out what a "fault-free" rectangle is and try to make some of your own.

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