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?
"I first saw it used as a homework question more than
ten years ago with a year seven class (11 year old students) in
Budapest and have used it with every group I have taught since -
initial teacher training, mathematics education research students,
in-service; everyone gets it. I love it because it forces an
acknowledgement of so many different topics in mathematics and is
sufficiently challenging to keep almost any group meaningfully
occupied within a framework of Key Stage Three mathematics content.
This, for me, is the key to a good problem; Key Stage Three content
alongside non-standard and unexpected outcomes. For example,
notwithstanding the obvious problem solving skills necessary for
managing such a non-standard problem, I think it requires an
understanding of coordinates, isosceles triangles and the area of a
triangle. It requires an awareness of the different factors of 18
and which are likely to yield productive solutions. It requires,
also, an understanding not only of basic transformations like
reflection and rotation but also an awareness of their symmetries.
Moreover, the solution, which is numerically quite small, is
attainable without being trivial. In short, I love this problem
because of the wealth of basic ideas it encapsulates and the sheer
joy it brings to problem solvers, of whatever age, when they see
why the answer has to be as it is. It is truly the best problem
ever and can provoke some interesting extensions."