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. . . .

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.

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 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.

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?

Relate these algebraic expressions to geometrical diagrams.

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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.

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.

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.

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.

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?

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

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

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

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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.

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

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

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

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

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.

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.

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.

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. . . .

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. . . .

Can you work out where the blue-and-red brick roads end?

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

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?

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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.

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. . . .

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

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.

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

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

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

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

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.