[1] The random close packing value is significantly below the maximum possible close-packing of same-size hard spheres into a regular crystalline arrangements, which is 74.04%.
[2] Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation.
The random close packing fraction of discs in the plane has also been considered a theoretically unsolved problem because of similar difficulties.
[3] The solution was found by limiting the probability of growth of ordered clusters to be exponentially small and relating it to the distribution of `cells', which are the smallest voids surrounded by connected discs.
If the objects are polydispersed then the volume fraction depends non-trivially on the size-distribution and can be arbitrarily close to 1.
[9] Products containing loosely packed items are often labeled with this message: 'Contents May Settle During Shipping'.