Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

This problem is designed to help children to learn, and to use, the two and three times tables.

Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

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.

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.

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

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

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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?

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Here are the prices for 1st and 2nd class mail within the UK. You have an unlimited number of each of these stamps. Which stamps would you need to post a parcel weighing 825g?

This article for teachers looks at how teachers can use problems from the NRICH site to help them teach division.

The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

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. . . .

Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

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

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.

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

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?

Are these statements always true, sometimes true or never true?

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

Grandma found her pie balanced on the scale with two weights and a quarter of a pie. So how heavy was each pie?

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

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

Chandrika was practising a long distance run. Can you work out how long the race was from the information?

Use the information to work out how many gifts there are in each pile.

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

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?

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

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

Can you each work out the number on your card? What do you notice? How could you sort the cards?

How would you count the number of fingers in these pictures?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

Resources to support understanding of multiplication and division through playing with number.

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?