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?

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?

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.

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?

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

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

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

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

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

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

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!

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.

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 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. . . .

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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

Keep constructing triangles in the incircle of the previous triangle. What happens?

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.

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.

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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

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

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

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

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?

When is it impossible to make number sandwiches?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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?

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

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?

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

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?

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

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

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

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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.