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

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.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

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

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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

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

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

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

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

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 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.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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.)

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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.

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?

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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?

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. . . .

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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...

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.

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.

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

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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

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 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.

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.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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.

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.

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

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?

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

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

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

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.