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

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

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.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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.

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.

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?

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

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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.

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.

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.

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

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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.

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

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

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?

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

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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

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

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

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.

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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?

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

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

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. . . .

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

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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

Which of these triangular jigsaws are impossible to finish?

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.