In Tomorrow's Math by Stanley Ogilvy (1972), it lists a minimal with polyhedra less than 19 faces as an unsolved problem. Wolfram currently lists that same example polyhedra and lists finding a more simple one as an unsolved problem:


According to the book (p77) "Richard K Guy, who devised this ingenious model and proved that it is unistable, proved also that no fewer than 17 lateral faces would do. In the same paper he also poses several new problems. In all cases density is assumed to be uniform:

  • Does there exist a unistable polyhedron of some pattern different from this one, with a total of fewer than 19 faces?
  • It is known that any tetrahedron is stable on at least two faces. A first step toward closing the gap between 4 and 19 would be to show that every pentahedrn is stable on at least 2 faces. Even this is not known.
  • Guy's 19-hedron has two axes of symmetry. Can a unistable polyhedron have more than two?
  • Can the resting face of a unistable convex polyhedron be the face of least diameter? ( In the 19-hedron it happens to be the face of greatest diameter).

My question is, is this original paper still available anywhere with Richard Guy's proofs? And second, is how would one might demonstrate unistability via a mathematics or by testing it built out in something like FreeCAD/Autocad?

Has there been any interesting progress on the other problems listed in the book?



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