This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

A huge wheel is rolling past your window. What do you see?

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

What fractions can you divide the diagonal of a square into by simple folding?

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Kyle and his teacher disagree about his test score - who is right?

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Prove Pythagoras' Theorem using enlargements and scale factors.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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. . . .

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!

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

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.

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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

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

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

Can you make sense of these three proofs of Pythagoras' Theorem?

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

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

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?