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?

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.

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

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

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

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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

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.

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. . . .

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.

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

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?

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?

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?

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

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.

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

On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

Can you score 100 by throwing rings on this board? Is there more than way to do it?

Where can you draw a line on a clock face so that the numbers on both sides have the same total?

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.

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

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

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

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

In November, Liz was interviewed for an article on a parents' website about learning times tables. Read the article here.

After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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

This task combines spatial awareness with addition and multiplication.

Have a go at balancing this equation. Can you find different ways of doing it?

Can you work out some different ways to balance this equation?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

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.

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

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

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

Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was.

Given the products of adjacent cells, can you complete this Sudoku?

Number problems at primary level that may require determination.

Number problems at primary level that require careful consideration.

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