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Shapely Figures (Posted on 2003-12-12) Difficulty: 3 of 5
What are the missing numbers in this sequence of ordered triplets?
{4, 6, ?}
{6, 12, ?}
{8, ?, 6}
{?, 30, 20}
{20, ?, 12}

See The Solution Submitted by SilverKnight    
Rating: 2.5000 (4 votes)

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Solution ... and of course | Comment 3 of 10 |
(In reply to Solution by Brian Smith)

By way of explanation, these are the regular (or Platonic) solids. There are of course irregular octahedra, dodecahedra and icosahedra (such as prisms with bases of (F-2) sides), and these are the figures for the regular ones.

Also, as the figures show, these solids have duals, the tetrahedron being its own dual, while the cube and octahedron are duals of each other as are the dodecahedron and icosahedron. The duals switch vertices for faces in each other. Thus the vertices column can be swapped with the faces column so long as the names of the cube and octahedron are switches as well as the dodecahedron and the icosahedron. Then the list is in order by number of vertices rather than number of faces--unusual only because of the naming conventions based on number of faces.
  Posted by Charlie on 2003-12-12 12:12:28

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