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?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Can you rearrange the cards to make a series of correct mathematical statements?
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
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!
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
When is it impossible to make number sandwiches?
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
If you think that mathematical proof is really clearcut and universal then you should read this article.
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.
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.
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.
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
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?
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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. . . .
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
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. . . .
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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.
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 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.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
Can you discover whether this is a fair game?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
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. . . .
Can you explain why a sequence of operations always gives you perfect squares?
Which of these roads will satisfy a Munchkin builder?
What fractions can you divide the diagonal of a square into by simple folding?
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
Can you use the diagram to prove the AM-GM inequality?
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?
Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?