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?

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. . . .

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

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

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.

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

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?

This task combines spatial awareness with addition and multiplication.

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

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

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

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?

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?

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?

This task offers an opportunity to explore all sorts of number relationships, but particularly multiplication.

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?

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

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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!

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

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

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?

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?

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

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

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

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

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.

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

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

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

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

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

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. . . .

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Number problems at primary level that may require resilience.

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

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

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,. . . .

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!

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

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

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.

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

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?

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.

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

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?