Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances. 
The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
 A: This is equivalent to sphere packing in a cube. This type of problem is messy. Even if you look at circle packings in a square, only a few configurations have been proved optimal. The best configurations known for many other values are complicated, and it's not easy to specify a short list of possible combinatorial types of configurations to test.
Hugo Pfoertner has tabulated numerical calculations for up to $72$ spheres in a cube. These are not guaranteed to be the best possible.
A: A very simple, cute "math-folklore" solution for $N=9$ goes as folows: the eight vertices and the center of the unit cube form a configuration of nine points, in which the minimum distance between them is $\sqrt3/2$. This is the maximum of the minimum over all configurations. Proof: partition the cube into eight cubes of edge length $1/2$. No matter how you place nine points in the unit cube, two of them must be contained in one of the small cubes; the distance between them is at most $\sqrt3/2$. With a bit more elaboration one can prove that the extreme configuration is unique. Analogous proofs work for five points in a square and seventeen points in the $4$-dim. cube, but fails in dimensions higher than $4$.
