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. . . .

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Find the values of the nine letters in the sum: FOOT + BALL = GAME

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?

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.

What is the sum of all the digits in all the integers from one to one million?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

You have four jugs of 9, 7, 4 and 2 litres capacity. The 9 litre jug is full of wine, the others are empty. Can you divide the wine into three equal quantities?

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

When I type a sequence of letters my calculator gives the product of all the numbers in the corresponding memories. What numbers should I store so that when I type 'ONE' it returns 1, and when I type. . . .

Try out some calculations. Are you surprised by the results?

A combination mechanism for a safe comprises thirty-two tumblers numbered from one to thirty-two in such a way that the numbers in each wheel total 132... Could you open the safe?

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

This article suggests some ways of making sense of calculations involving positive and negative numbers.

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

How can we help students make sense of addition and subtraction of negative numbers?

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. . . .

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

Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.

Got It game for an adult and child. How can you play so that you know you will always win?

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

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

Different combinations of the weights available allow you to make different totals. Which totals can you make?

If you wrote all the possible four digit numbers made by using each of the digits 2, 4, 5, 7 once, what would they add up to?

An account of some magic squares and their properties and and how to construct them for yourself.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was. . . .

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

Find out about Magic Squares in this article written for students. Why are they magic?!

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Choose any three by three square of dates on a calendar page...

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Delight your friends with this cunning trick! Can you explain how it works?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?