Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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 $F_{n+1}F_{n-1}$.

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.

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

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.

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?

An inequality involving integrals of squares of functions.

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

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

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!

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

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.

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

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

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

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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?

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.

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

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

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

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

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

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

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.

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.

When is it impossible to make number sandwiches?

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

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

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?

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. . . .

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?

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

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.

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

An introduction to some beautiful results of Number Theory

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

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