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

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

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

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.

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

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

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

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?

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.

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.

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.

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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?

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!

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?

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. . . .

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?

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 these three proofs of Pythagoras' Theorem?

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

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.

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

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

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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

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

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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 Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

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

Kyle and his teacher disagree about his test score - who is right?

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.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.