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

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?

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?

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

Can you explain the strategy for winning this game with any target?

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

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?

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?

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

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

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

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?

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

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.

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

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

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....?

Use the interactivities to complete these Venn diagrams.

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?

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

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

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

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?

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?

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?

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?

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

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

An interactive activity for one to experiment with a tricky tessellation

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

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?

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. . . .

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.

Can you complete this jigsaw of the multiplication square?

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

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?

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

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?

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

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?