# A D20 (dice) has sides where $N-1,N,N+1$ are always neighbours on surface of solid. For which DN is this possible?

A regular gon D20 dice used for example in various forms of gambling and trading card games is shown below

As can be seen each number $$N$$ residing on some face has two of it's neighbouring faces with numbers $$N-1$$ and $$N+1$$. Can we prove somehow for which regular N-gons this is possible to do?

• What do you mean by always neighbours? In your picture, what makes $18$ and $20$ neighbours? For reasonable definitions, it looks probable to me that all five regular polyhedra can have this labelling – Henry May 14 at 10:12
• @Henry always exist two of the "face" neighbours (neighbours in sense of sharing an edge) for each face so that one of these face neighbours has number which is 1 smaller and the other 1 larger than the value of the current face. – mathreadler May 14 at 10:19
• In that case it looks easy just to show a labelling that works for each of the five cases (effectively a Hamiltonian path on the dual graph). Slightly harder, at least for the dodecahedron, would be requiring additionally all three successive faces to also share a vertex, but I think this too is possible – Henry May 14 at 10:26
• Wait you mean so that they circle around surface like a snake? – mathreadler May 14 at 10:30
• That is the way I would do it – Henry May 14 at 10:59