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?

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

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

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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. . . .

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

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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. . . .

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

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

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.

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

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

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. . . .

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

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

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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. . . .

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

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

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!

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

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.

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

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

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. . . .

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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

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 shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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.

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.

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?

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?

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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?