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

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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. . . .

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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. . . .

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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

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.

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

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

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

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

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.

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.

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.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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

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.

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.

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

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 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?

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

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

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

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

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.

Have a go at being mathematically negative, by negating these 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?

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

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

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.

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

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?

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.