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

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

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!

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

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?

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

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?

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

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.

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

Can you complete this jigsaw of the multiplication square?

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?

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?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

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

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?

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

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?

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

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

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

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

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.

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

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.

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?

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

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

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?

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

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

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.

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?

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

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.

An interactive activity for one to experiment with a tricky tessellation