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

What can you say about the common difference of an AP where every term is prime?

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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. . . .

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

By proving these particular identities, prove the existence of general cases.

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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.

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

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.

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.

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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!

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. . . .

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

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.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

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

Prove Pythagoras' Theorem using enlargements and scale factors.

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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. . . .

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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

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?

Have a go at being mathematically negative, by negating these statements.

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.

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

Do you have enough information to work out the area of the shaded quadrilateral?

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

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