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

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

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.

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.

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 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?

Prove Pythagoras' Theorem using enlargements and scale factors.

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?

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.

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

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

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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?

Four jewellers share their stock. Can you work out the relative values of their gems?

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

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.

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.

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

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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.

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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. . . .

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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

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

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. . . .

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!

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

Do you have enough information to work out the area of the shaded quadrilateral?

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.

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?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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?

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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

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

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

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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?

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

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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.