@Douglas: I would guess the question means: among those convex icosahedra whose dihedral angles are all equal, which one maximises this dihedral angle?

Better still, since there is trivially a longest common subsequence, say "among the longest common subsequences, there is at least one which is a palindrome".

@Benoit: I'm not sure it's very sensible to assume that the polyhedron will land on the face above which the c.o.g. lies, if we understand "land" to mean "come to rest": it's easy to construct a polyhedron which has a face which subtends positive solid angle at the c.o.g., but such that the die is unstable on that face.

@Andre: That would be nicer, but extremely misleading. The Riemann hypothesis is an interesting question which is still unresolved; the question of a non-isohedral fair die is either so ill-posed as to render the term "unresolved" meaningless, or so very specialised (as in the model proposed in Joseph's answer) to call into question the claim that it is interesting.

@Joseph: I'm sorry if I was rather dismissive of your original question; I was really venting in response to non-mathematicians who say things like "there must be an answer; why can't you mathematicians work it out?". What you've proposed here is a sensible (though probably unrealistic) model. But I suspect that it will be extremely difficult to analyse, except maybe in the case of zero elasticity.