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

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?

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

This article for teachers suggests ideas for activities built around 10 and 2010.

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

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

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

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?

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

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?

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 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?

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. . . .

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?

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?

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.

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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.

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

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?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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

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

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.

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.

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

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

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

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

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

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

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

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!

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?

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?

Can you complete this jigsaw of the multiplication square?

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

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

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

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

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?

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.

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.

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

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

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