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This problem looks at how one example of your choice can show something about the general structure of multiplication.

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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?

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Play this game and see if you can figure out the computer's chosen number.

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

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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.

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How will you work out which numbers have been used to create this multiplication square?

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This challenge combines addition, multiplication, perseverance and even proof.

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This task combines spatial awareness with addition and multiplication.

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

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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.

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

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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?

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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?

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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.

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What is the smallest number of answers you need to reveal in order to work out the missing headers?

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There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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

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

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

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This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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

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In this simulation of a balance, you can drag numbers and parts of number sentences on to the trays. Have a play!

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

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

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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?

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I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

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Can you find different ways of creating paths using these paving slabs?

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

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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?

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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?

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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?

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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?

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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?

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

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Can you put these four calculations into order of difficulty? How did you decide?

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

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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?

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

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

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Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

In this article for primary teachers, Lynne McClure outlines what is meant by fluency in the context of number and explains how our selection of NRICH tasks can help.

This article for primary teachers encourages exploration of two fundamental ideas, exchange and 'unitising', which will help children become more fluent when calculating.

In this article, Alf outlines six activities using the Gattegno chart, which help to develop understanding of place value, multiplication and division.

Alf describes how the Gattegno chart helped a class of 7-9 year olds gain an awareness of place value and of the inverse relationship between multiplication and division.

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

Related resources supporting pupils' understanding of multiplication and division through playing with numbers.

More resources to support understanding multiplication and division through playing with numbers