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?

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!

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?

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

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.

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

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

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?

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

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?

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

Can you complete this jigsaw of the multiplication square?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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

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

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

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?

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

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

An environment which simulates working with Cuisenaire rods.

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?

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

An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.

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

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?

An interactive activity for one to experiment with a tricky tessellation

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

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?

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

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

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

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

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. . . .

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

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?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

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?

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

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 create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards?

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?

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.

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

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