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

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

What happens when you try and fit the triomino pieces into these two grids?

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 a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

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

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

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

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....?

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

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

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?

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

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?

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

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?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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

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

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

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

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

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

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

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

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

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

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

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?

Try out the lottery that is played in a far-away land. What is the chance of winning?

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

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?

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

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

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?

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?

Sort the houses in my street into different groups. Can you do it in any other ways?

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 your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

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?

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

Can you complete this jigsaw of the multiplication square?

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