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

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. . . .

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?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

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

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.

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.

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?

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

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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

Can you find the areas of the trapezia in this sequence?

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?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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

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

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

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

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

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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.

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

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.

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.

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.

Prove Pythagoras' Theorem using enlargements and scale factors.

Four jewellers share their stock. Can you work out the relative values of their gems?

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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. . . .

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?

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.

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.

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.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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

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

When is it impossible to make number sandwiches?

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

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!

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?

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

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?

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