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.

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.

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

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. . . .

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.)

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!

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

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.

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.

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.

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

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 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. . . .

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

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

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

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.

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?

Which of these triangular jigsaws are impossible to finish?

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

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?

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

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

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

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

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?

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 orf two or more cubes.

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

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

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

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

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.

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

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

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. . . .

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. . . .

An inequality involving integrals of squares of functions.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.