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.

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

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

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

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?

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

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

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.

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 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.

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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.

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

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. . . .

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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

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

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.

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

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.

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

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.

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.

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.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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

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.

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

When is it impossible to make number sandwiches?

By proving these particular identities, prove the existence of general cases.

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?

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

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

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

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

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

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

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

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?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Prove Pythagoras' Theorem using enlargements and scale factors.

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.