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.

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. . . .

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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?

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. . . .

The value of the circle changes in each of the following problems. Can you discover its value in each problem?

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

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?

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

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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.

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

56 406 is the product of two consecutive numbers. What are these two numbers?

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?

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.

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.

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?

What is the least square number which commences with six two's?

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

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Number problems at primary level that may require resilience.

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

This task combines spatial awareness with addition and multiplication.

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?

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?

What is the remainder when 2^{164}is divided by 7?

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?

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?

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?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

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

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

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

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

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

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!

This task offers an opportunity to explore all sorts of number relationships, but particularly multiplication.

Number problems at primary level that require careful consideration.

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

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?

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

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?

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