Creating and manipulating expressions and formulae

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

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.

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

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?

If $a \times b=2$, $b \times c=24$, $c \times a=3$ and $a$, $b$ and $c$ are positive, what is the value of $a+b+c$?

Given that the number 2008 is the correct answer to a sum, can you find n?

The mean of three numbers x, y and z is x. What is the mean of y and z?

Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?