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?

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. . . .

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.

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

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

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

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.

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?

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

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

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!

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.

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?

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

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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

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.

A huge wheel is rolling past your window. What do you see?

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

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. . . .

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

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.

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

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

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?

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

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?

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?

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?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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

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.

Can you explain why a sequence of operations always gives you perfect squares?

When is it impossible to make number sandwiches?

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

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?