Without loss of generality, let the smallest distance be $ 1$. We claim that the greatest distance must be at least $ 9$. To show this, assume the greatest distance was $ <9$. This means that the entire set of points would fit into a sphere of radius $ 4.5$. Note however that each pair of points is at least $ 1$ apart; thus there are no points within a sphere of radius $ 1$ of a given point - if we can show that we cannot fit all of these inside a sphere of radius $ 5.5$, we are done. Each sphere intersects at most 12 other spheres, so if we remove $ \frac{11}{12}$ of them, we can attain a set of at least $ 167$ disjoint spheres. But the sum of the volumes of these is greater than the volume of a sphere with radius $ 5.5$, so we are done.