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

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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.

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.

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

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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

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.

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.

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

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?

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

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

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!

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.

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.

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

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

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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. . . .

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

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Can you make sense of the three methods to work out the area of the kite in the square?

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.

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?

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.

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.

What fractions can you divide the diagonal of a square into by simple folding?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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. . . .

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

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

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

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

Can you make sense of these three proofs of Pythagoras' Theorem?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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.

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

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .