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.

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.

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

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

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.

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

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

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

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .

An inequality involving integrals of squares of functions.

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.

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.

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.

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 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.

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

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.

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...

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

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

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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

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

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

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

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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

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

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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?

Take any whole number q. Calculate q^2 - 1. Factorize q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .

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.

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

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

Explore a number pattern which has the same symmetries in different bases.

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.

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.

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.

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

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?

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.

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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?

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 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.