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 A New Solid (Posted on 2004-10-05)
A regular tetrahedron has four equilateral triangles as faces. A regular square pyramid has four equilateral triangles and a square as faces. The faces of the tetrahedron are congruent to the triangular faces of the square pyramid.

A new polyhedron is created by gluing the tetrahedron and the square pyramid together at a triangular face so that the vertices of the triangles coincide. How many faces does this polyhedron have?

 Submitted by Brian Smith Rating: 4.1250 (8 votes) Solution: (Hide) At first one might guess 7 faces as 2 faces vanish. But the polyhedron has only 5 faces as two rhombics are created. To see this place two pyramids as shown in the figure. (ABCDX and CDEFY) A D E X Y B C F We know CX=DX=CD=CY=DY. Since CDEFY is a translation of ABCDX, then XY=CD and CDXY is a regular tetrahedron. The new polyhedron is formed by ABCDXY. Since AD is parallel to XY, ADXY is a single face. Similarily BCXY is a single face. The other three faces of this solid are ABCD, ABX, and CDY. The new polyhedron has five faces total.

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 Subject Author Date Answer K Sengupta 2008-11-25 00:06:01 solution daniel 2004-12-18 01:10:18 re(2): Amazing coincidence Ken Haley 2004-10-08 00:09:30 re: Amazing coincidence nikki 2004-10-06 08:12:03 good one rixar 2004-10-06 07:50:23 re(2): Surely not the intended solution ! (spoilers) SilverKnight 2004-10-06 03:52:53 Amazing coincidence Ken Haley 2004-10-06 00:28:20 re: Perhaps simpler calculations Ken Haley 2004-10-06 00:20:29 re: Perhaps simpler calculations David Shin 2004-10-05 20:36:55 re: Very Simply Neat Charlie 2004-10-05 20:25:48 Very Simply Neat CeeAnne 2004-10-05 19:15:47 Perhaps simpler calculations Tristan 2004-10-05 18:56:19 Link with Figures Richard 2004-10-05 14:02:51 Another way of showing solution Charlie 2004-10-05 13:52:03 re: Surely not the intended solution ! (spoilers) nikki 2004-10-05 13:10:14 re(2): Surely not the intended solution ! Charlie 2004-10-05 13:05:24 re: Surely not the intended solution ! Jonathan Fletcher 2004-10-05 12:38:22 Solution David Shin 2004-10-05 12:22:56 Surely not the intended solution ! Syzygy 2004-10-05 12:08:24

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