Wodzimierz Kuperburg - Covering the cube with equal balls

Define c^d(k), the covering radius, as the minimum radius of k congruent balls that can cover the unit d-dimensional cube. (The packing radius is defined similarly.) In general, the problem of determining the values of c^d(k) as well as the corresponding packing problem seem to be extremely difficult even in dimension 2. On the packing problem, several results have been obtained in dimension 2by various authors, and a few results in dimension 3, mainly by J. Schaer. We discuss the covering problem and we determine c^d(2), c³(3), c³(4), c³(8), c⁴(4), and c⁴(16) along with the optimal configurations of balls that produce them. Also, we state conjectures on the remaining values of c³(k) and their ball configurations for .