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

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?

  • $\begingroup$ 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 $\endgroup$ – Henry May 14 at 10:12
  • $\begingroup$ @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. $\endgroup$ – mathreadler May 14 at 10:19
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    $\begingroup$ 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 $\endgroup$ – Henry May 14 at 10:26
  • $\begingroup$ Wait you mean so that they circle around surface like a snake? $\endgroup$ – mathreadler May 14 at 10:30
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    $\begingroup$ That is the way I would do it $\endgroup$ – Henry May 14 at 10:59

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