Creating and manipulating expressions and formulae

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 metres from B. How long is the lake?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Kyle and his teacher disagree about his test score - who is right?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Can you produce convincing arguments that a selection of statements about numbers are true?

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

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?

Granny has taken up deep-sea fishing! Last week, she caught a fish so big that she had to cut it into three pieces in order to weigh it...

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 $a+b+c$?