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

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

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.

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.

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

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

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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

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.

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

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

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.

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

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

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.)

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

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

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

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.

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.

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!

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.

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

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

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

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. . . .

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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

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. . . .

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.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

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.

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?

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

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.

An article which gives an account of some properties of magic squares.

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

Can you work out where the blue-and-red brick roads end?

Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing

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.

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.