Freddie Manners, age 11 from Packwood Haugh School, Shropshire sent
in the following beautiful solution. Freddie asks ``Is this
relationship to the Golden Ratio coincidental?'' The answer is
probably not. Mathematics if full of connections which at first
seem surprising. The question involves the sides of a right-angled
triangle, the cube of the Golden Ratio $\varphi = {1\over
2}(1+\sqrt{5})$, and the arithmetic, geometric and harmonic means
of two number (AM, GM and HM respectively). Firstly Freddie found
the cube of $\varphi = {1\over 2}(1+\sqrt{5})$.