Maximum Scattering

Having done the first two parts of the question you can show similarly that, with $n$ numbers, there exists a certain value of the variance and hence that the variance can be at least that large. It is quite a subtle point, but you can't be sure that you have found the largest value of the variance with $n = 4$ or more without assuming that the geometrical reasoning generalises from $3$ dimensions to higher dimensions. The results are in fact valid in $n$-dimensional geometry but you are not asked to prove this.