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

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.

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 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. . . .

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. . . .

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. . . .

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.

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

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

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

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

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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

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?

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. . . .

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

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. . . .

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 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.

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

Relate these algebraic expressions to geometrical diagrams.

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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.

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?

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

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.

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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

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

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

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.

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.

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

When is it impossible to make number sandwiches?

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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

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

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

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

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