The diameter 5 problem has (ignoring rotations and reflections) http://www.comm.toronto.edu/~frank/hexagon.html an unique solution
10,12,16 13,4,2,19 15,8,5,7,3 14,6,1,17 9,11,18
There is also a (trivial) diameter 1 magic hexagon. These are the only two sizes of magic hexagon possible.
Eliminating symmetry is important. The diameter 5 problem has 12 symmetries (rotation, reflection, …) which need to be considered.