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

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?

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?

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?

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

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!

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

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?

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

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

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?

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

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

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?

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....?

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 make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

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

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

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 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?

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

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

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?

An interactive activity for one to experiment with a tricky tessellation

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?

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

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

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

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

Board Block game for two. Can you stop your partner from being able to make a shape on the board?

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?

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

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

Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?

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

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 tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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

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?