Well, you can have at most two vertices with an odd number of edges meeting there because if you have a vertex like that you have to either start or end there. So that rules out the cube, dodecahedron, icosahedron and tetrahedron. So of the platonic solids, only the octahedron is possible. And that's easy, start at any point, trace one of the equatorial squares meeting at that vertex, then move one edge along the other equatorial square, trace the equatorial square at right angles to it, then complete the remaining three edges of that equatorial square, and you are done.
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Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
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