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. . . .

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

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.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

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.

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

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

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. . . .

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

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.

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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 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. . . .

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.

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...

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. . . .

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?

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

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

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

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

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 make sense of these three proofs of Pythagoras' Theorem?

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

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

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!

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

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

What fractions can you divide the diagonal of a square into by simple folding?

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.

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

A introduction to how patterns can be deceiving, and what is and is not a proof.

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.

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?

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

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

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?

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

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.

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?

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