Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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

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

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.

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?

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.

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?

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?

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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.

When is it impossible to make number sandwiches?

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

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!

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

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.

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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.

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

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.

What fractions can you divide the diagonal of a square into by simple folding?

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

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. . . .

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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?

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

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

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

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

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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?

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. . . .

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

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.

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

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

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

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?

Can you make sense of the three methods to work out the area of the kite in the square?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?