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

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

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

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

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

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

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?

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?

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

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

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

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

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

A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins.

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....?

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

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

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?

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

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

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

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?

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

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?

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

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

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?

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

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?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

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?

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?

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

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

Take it in turns to make a triangle on the pegboard. Can you block your opponent?

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

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?