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

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

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

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

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?

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

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

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?

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?

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?

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?

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!

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?

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

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

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

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?

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

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 explain the strategy for winning this game with any target?

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....?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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

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

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

Exchange the positions of the two sets of counters in the least possible number of moves

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

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 fit the tangram pieces into the outline of Granma T?

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

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

Train game for an adult and child. Who will be the first to make the train?

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outlines of these clocks?