Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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?

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.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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 the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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?

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

How many winning lines can you make in a three-dimensional version of noughts and crosses?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

Derive an equation which describes satellite dynamics.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which relates triangular numbers to square numbers?

Can you find a rule which connects consecutive triangular numbers?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Five equations... five unknowns... can you solve the system?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?