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

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

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

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. . . .

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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. . . .

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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. . . .

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

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.

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

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

When is it impossible to make number sandwiches?

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 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.

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!

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

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

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

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. . . .

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

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

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. . . .

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.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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?

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

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}.

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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?

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

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

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

Can you make sense of the three methods to work out the area of the kite in the 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?

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

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?

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