$n$ points are placed on an $n-2$-sphere so that the smallest angle from the centre between any pair of the points is maximised. What is this smallest angle?
$n=1 \ \ \ \cos^{-1}{1}\\ n=2 \ \ \ \cos^{-1}{-1}\\ n=3 \ \ \ \cos^{-1}{\frac{-1}2}\\ n=4 \ \ \ \cos^{-1}{\frac{-1}3}$
The smallest angle will be the same for any $n$ points on a $p$-sphere where $p> n-2$ as when $p=n-2$.