Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

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

Choose any three by three square of dates on a calendar page...

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

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

This is the second article on right-angled triangles whose edge lengths are whole numbers.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

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. . . .

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

When is it impossible to make number sandwiches?

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

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Here are some examples of 'cons', and see if you can figure out where the trick is.