Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about. . . .

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Can you find any perfect numbers? Read this article to find out more...

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

What is the remainder when 2^{164}is divided by 7?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

This number has 903 digits. What is the sum of all 903 digits?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

This activity creates an opportunity to explore all kinds of number-related patterns.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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?

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

Using only the red and white rods, how many different ways are there to make up the other rods?

Here are some ideas to try in the classroom for using counters to investigate number patterns.

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

What are the coordinates of this shape after it has been transformed in the ways described? Compare these with the original coordinates. What do you notice about the numbers?

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.