You may also like

Spread

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

David and Goliath

Does weight confer an advantage to shot putters?

Maximum Scattering

Age 16 to 18
Challenge Level

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.