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?

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

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

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!

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

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

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 explain the strategy for winning this game with any target?

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?

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.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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?

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?

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?

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

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

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

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?

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

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 make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

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?

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

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

Carry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.

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?

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

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

Imagine picking up a bow and some arrows and attempting to hit the target a few times. Can you work out the settings for the sight that give you the best chance of gaining a high score?

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

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

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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?

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?