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

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?

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?

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

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.

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

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

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?

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...

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

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. . . .

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

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.

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

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

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.

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...

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

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

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

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

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. . . .

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. . . .

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

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. . . .

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

Can you explain why a sequence of operations always gives you perfect squares?

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.

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

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

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

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.

Can you make sense of these three proofs of Pythagoras' Theorem?

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

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

When is it impossible to make number sandwiches?

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?

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

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

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. . . .

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

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

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.