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.

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.

There are over sixty different ways of making 24 by adding, subtracting, multiplying and dividing all four numbers 4, 6, 6 and 8 (using each number only once). How many can you find?

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?

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?

Can you find what the last two digits of the number $4^{1999}$ are?

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

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

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?

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

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.

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?

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 clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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. . . .

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Can you replace the letters with numbers? Is there only one solution in each case?

Find a great variety of ways of asking questions which make 8.

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

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?

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?

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?

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 challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

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

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

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

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

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

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

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 task combines spatial awareness with addition and multiplication.

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

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

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?

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

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.

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

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

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!

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 challenge combines addition, multiplication, perseverance and even proof.

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?

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