Creating and manipulating expressions and formulae

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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.

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

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 before being reflected across to the opposite wall and so on until it hits the screen.

The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?

Can you explain how this card trick works?

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 lines are 1 unit and 2 units.