Puzzle #131: The Pyramid Problem

This is a beautiful geometry puzzle contributed by my nephew, Karan Sharma. The puzzle originally appeared in a PSAT exam in 1980, and was not really a “puzzle” until one student proved that there was a different answer than what everyone had thought. Enough in already!

Here it goes – A tetrahedron and a pyramid with edges of unit length are glued together at one triangular face. How many exposed faces does the resulting solid have?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

Happy geometry day!

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3 Responses to Puzzle #131: The Pyramid Problem

  1. Abhinav Jain says:

    5 faces

  2. 5 faces.

    Borrow another pyramid. Keep pyramids side by side keeping one edge of the square in common. You will see that the space in between can be filled by a tetrahedron – with the edges as the common base of the the two pyramids, two sides of the triangle on pyramid1, two sides of the triangle on pyramid2, the edge joining the tips of pyramid.

    This almost creates a trapezium type thing. This proves that two faces of tetrahedron would really be in the same plane as two faces of pyramid, and hence, after the two are kept on top of each other – number of faces would be 4+5-2-2=5

  3. Suman Saraf says:

    This was a beautiful problem but ruined by the excessive hints IMHO 🙂 If it was asked in a test, I would have probably marked 7 though when I read your version of the problem, it was obvious some faces had to be co-planar. With a bit of visualization, you can see that the answer should be 5.

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