Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

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

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

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. . . .

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

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

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?

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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.

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

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. . . .

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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

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?

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

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. . . .

A huge wheel is rolling past your window. What do you see?

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

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

Keep constructing triangles in the incircle of the previous triangle. What happens?

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.

How many noughts are at the end of these giant numbers?

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. . . .

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?

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.

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}.

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

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

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. . . .

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.

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.

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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

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.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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

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?

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

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