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?

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.

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

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .

Which set of numbers that add to 10 have the largest product?

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?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

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. . . .

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?

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

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

These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?

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

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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

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?

Number problems at primary level that may require resilience.

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?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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,. . . .

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?

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

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.

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

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?

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .

This Sudoku requires you to do some working backwards before working forwards.

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

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.

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

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?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

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

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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

This task combines spatial awareness with addition and multiplication.

Here is a chance to play a fractions version of the classic Countdown Game.

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

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

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.

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?