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

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

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Can you work through these direct proofs, using our interactive proof sorters?

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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

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

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.

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.

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

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.

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.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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

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!

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?

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

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.

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.

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

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

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?

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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.

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.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Sort these mathematical propositions into a series of 8 correct statements.

When is it impossible to make number sandwiches?

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

A introduction to how patterns can be deceiving, and what is and is not a proof.

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

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

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?

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

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

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

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

Can you rearrange the cards to make a series of correct mathematical statements?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?