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. . . .

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.

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.

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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.

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

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?

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.

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.

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

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

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

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

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

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

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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

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

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. . . .

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

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

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Prove Pythagoras' Theorem using enlargements and scale factors.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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

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

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?

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?

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

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

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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

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

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

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

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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?

Here are some examples of 'cons', and see if you can figure out where the trick is.