Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you work out the area of the inner square and give an
explanation of how you did it?
David of Madras College, St Andrew's had a
novel way of looking at this. He first rearranged the semicircles
into three circles touching at $A$. He then pointed out that, when
working out the areas, all the answers will be multiples of $\pi$
and ultimately, when comparing the areas, it is necessary to
factorise out $\pi$ anyway. David's geometrical representation of
the results "when divided by $\pi$'', compares the equivalent areas
Thomas of Simon Langton School, Canterbury
also sent in a good solution.