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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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?
Some diagrammatic 'proofs' of algebraic identities and inequalities.
An article which gives an account of some properties of magic squares.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
Can you rearrange the cards to make a series of correct mathematical statements?
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.
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.
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.
If you think that mathematical proof is really clearcut and universal then you should read this article.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Which of these roads will satisfy a Munchkin builder?
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?
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.
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. . . .
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
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. . . .
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Kyle and his teacher disagree about his test score - who is right?
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
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.
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?
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.
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. . . .
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
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.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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?
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.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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.