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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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.

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

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

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

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. . . .

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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

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

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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. . . .

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?

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

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

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. . . .

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

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

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. . . .

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?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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

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

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

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

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

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?

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

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.

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.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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...

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. . . .

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

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.

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.

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

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

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.

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.

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!

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