In this article for teachers, Elizabeth Carruthers and Maulfry Worthington explore the differences between 'recording mathematics' and 'representing mathematical thinking'.

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

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

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

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?

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

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

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

On Friday the magic plant was only 2 centimetres tall. Every day it doubled its height. How tall was it on Monday?

Number problems at primary level that may require determination.

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?

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

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?

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

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

Can you each work out the number on your card? What do you notice? How could you sort the cards?

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.

Number problems at primary level that require careful consideration.

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

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?

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 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.

Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?

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?

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?

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?

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.

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

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

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?

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

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

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.

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

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

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

Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece?

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

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

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

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?

Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?

This task combines spatial awareness with addition and multiplication.

Are these statements always true, sometimes true or never true?

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