Formulae/strategy for performance of deductive reasoning on a Platonic solid?

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I have a Platonic solid (in this case, a dodecahedron), and I have twelve names that I must put on this dodecahedron. I have an incomplete list of who is next to who, and I'm trying to fill in the rest based on context clues. It seems like this would be a perfect problem for deductive reasoning, but the methods I learned in school are decidedly two-dimensional. Are there any formulas or techniques for applying this concept to a 3D shape?

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Would using the Schlegel diagram for of a dodecahedron help to $2$ dimensionalise it for you ? https://en.wikipedia.org/wiki/Schlegel_diagram#Examples

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