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.

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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?

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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

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

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. . . .

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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

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?

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

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

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?

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

Bernard Bagnall recommends some primary school problems which use numbers from the environment around us, from clocks to house numbers.

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?

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.

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?

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?

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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

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

Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

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

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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!

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

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

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

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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

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?

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

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?

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?

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

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

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

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

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.