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?
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.
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?
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.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
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.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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. . . .
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Can you rearrange the cards to make a series of correct mathematical statements?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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!
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
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.
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?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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.
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.
Keep constructing triangles in the incircle of the previous triangle. What happens?
Four jewellers share their stock. Can you work out the relative values of their gems?
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?
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. . . .
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
If you think that mathematical proof is really clearcut and universal then you should read this article.
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. . . .
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
When is it impossible to make number sandwiches?
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. . . .
An article which gives an account of some properties of magic squares.
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
What fractions can you divide the diagonal of a square into by simple folding?
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?
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. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
Prove Pythagoras' Theorem using enlargements and scale factors.