Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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. . . .

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

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

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?

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?

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

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

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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.

Relate these algebraic expressions to geometrical diagrams.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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

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.

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. . . .

If you think that mathematical proof is really clearcut and universal then you should read this article.

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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

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.

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

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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

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

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

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.

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

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. . . .

Can you find the value of this function involving algebraic fractions for x=2000?

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

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.

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

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

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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.

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

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?

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

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.

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .