challenge problem - - edges of a polyhedron

[lookagain]

I picked too hard of a problem , so I deleted it.

 

[lookagain]

Beyond my humble capabilities; but can you not do this:

1: cut a cube in half, the cut along a diagonal

2: on one of the resulting triangular surfaces,
build a triangular based pyramid

I now "see" 7 > > surfaces < < (bottom one being the 2nd triangular surface)

I'm rushing to the corner store to buy a cube of cheese....

"surfaces?" \(\displaystyle \ \ \ \ \)Are you making a reference/comparison to faces of a polyhedron? \(\displaystyle \ \ \ \) If so, I'm not sure why you are,
because I am asking about seven edges being possible or not. \(\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)

Or, are you intending to tie in the notion of the number of your "surfaces" to the number of edges of which I am asking?