A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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}.

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

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

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.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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. . . .

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

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. . . .

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. . . .

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Kyle and his teacher disagree about his test score - who is right?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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?

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

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

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

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

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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?

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

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

If you think that mathematical proof is really clearcut and universal then you should read this article.

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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

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

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

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?

When is it impossible to make number sandwiches?

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

Can you find the areas of the trapezia in this sequence?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.