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

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

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

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.

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?

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?

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?

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?

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. . . .

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. . . .

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.

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.

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

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

In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?

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. . . .

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

There are nasty versions of this dice game but we'll start with the nice ones...

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Find a great variety of ways of asking questions which make 8.

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 at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100.

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.

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.

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

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. . . .

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?

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

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.

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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

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.

A game for 2 or more players with a pack of cards. Practise your skills of addition, subtraction, multiplication and division to hit the target score.

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

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?

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

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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

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

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

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?