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

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

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

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

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. . . .

Find the sum of all three-digit numbers each of whose digits is odd.

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

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

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?

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.

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 task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

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

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

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

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?

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

This Sudoku, based on differences. Using the one clue number can you find the solution?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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

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?

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

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

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

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Try out this number trick. What happens with different starting numbers? What do you notice?

Can you find different ways of creating paths using these paving slabs?

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

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

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

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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?

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.

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

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

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?

Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest?

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

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?

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?

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?