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

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

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

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.

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

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.

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?

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.

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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

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

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

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?

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

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

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

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

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.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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 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?

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

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.

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

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.

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

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.

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?

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.

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.

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

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.

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

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.

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. . . .

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

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.

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

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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

When is it impossible to make number sandwiches?