From Michael Branicky comes a challenge involve many, many dice:

I have five kinds of fair Platonic dice: tetrahedra (whose faces are numbered 1-4), cubes (numbered 1-6), octahedra (numbered 1-8), dodecahedra (numbered 1-12) and icosahedra (numbered 1-20).²

When I roll two of the cubes, there is a single most likely sum: seven. But when I roll one cube and two tetrahedra, there is no single most likely sum eight and nine are both equally likely.

Which whole numbers are never the single most likely sum, no matter which combinations of dice I pick?