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Solve the Icosian Game

Find the Hamiltonian cycle along the edges of the dodecahedron.

    
First, find the net representation of a polyhedron from built-in data:
Wolfram Language code: data[property_] := PolyhedronData["Dodecahedron", property];
Wolfram Language code: faceStyle = Directive[ColorData["Rainbow"][0.25], Opacity[.7], EdgeForm[LightGray]];
Wolfram Language code: Row[{Graphics3D[{faceStyle, data["GraphicsComplex"]}], Spacer[35], g = data["SkeletonGraph"]}]
Now visualize the Hamiltonian cycle:
Wolfram Language code: h = PathGraph[First[FindHamiltonianCycle[g]]];
Wolfram Language code: tubeStyle = Directive[Red, Opacity[1]];
Wolfram Language code: Row[{Graphics3D[{faceStyle, data["GraphicsComplex"], tubeStyle, Tube[data["Vertices"][[Append[#, #[[1]]]&@VertexList[h]]], 0.1]}], Spacer[35], HighlightGraph[g, h, GraphHighlightStyle -> "Thick"]}]