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

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

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

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.

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.

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

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

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.

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

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

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

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?

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?

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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.

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

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

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

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.

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.

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

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

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

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.

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.

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.

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.

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

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?

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

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!

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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

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

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?

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

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

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

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

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. . . .

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?