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?

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

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.

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.

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?

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.

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

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

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

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

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

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.

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?

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?

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

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.

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

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

Can you explain why a sequence of operations always gives you perfect squares?

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

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!

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

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.

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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?

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?

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Four jewellers share their stock. Can you work out the relative values of their gems?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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

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

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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

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.

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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

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.

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.