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

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 visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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.

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

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?

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?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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.

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

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. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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. . . .

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 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. . . .

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

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.

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

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

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

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. . . .

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

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

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

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

How many noughts are at the end of these giant numbers?

When is it impossible to make number sandwiches?

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. . . .

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.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

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?

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

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

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!

Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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?

Can you explain why a sequence of operations always gives you perfect squares?

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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?

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.