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?

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

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

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!

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....?

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?

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

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?

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?

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

An interactive activity for one to experiment with a tricky tessellation

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

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

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

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

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!

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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?

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?

An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .

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?

A game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking.

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

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

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

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

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

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?

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?

Can you complete this jigsaw of the multiplication square?

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?

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

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

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

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

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 locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

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

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?

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

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 rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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