Simple geometry. Or is it?

I've got a regular tetrahedron and a square pyramid. Every edge of the two solids has the same length. If I perfectly attach one face of the tetrahedron to one of the triangular faces of the square pyramid (I.E. every point of one face overlaps a point of the other face, edges and vertexes included) how many faces will the new solid have?

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Edit, let me clarify: this only has to do with geometry. No lateral-thinking, no word puns, no silly explanations (otherwise I would have added some of these tags), just pure and simple geometry. Yes, it may look stupid, but it isn't.

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Inspired by "How many faces does the resulting polyhedron have?" on Math SE.

Answer

The answer is

5

I can't draw this, but

if the square pyramid sits on its base
and has its four triangular faces oriented N S E W,
and you set the tetrahedron to the east of the pyramid on its base,
with one side facing full west - call this face 1,
leaving face 2 pointing roughly NE and face 3 facing roughly SE,
and face 4 the face it sits on the ground on...

now tip the tetrahedron over to rest its face 1 against the E face of the pyramid, and glue it like that. you'll find that the edge between its faces 2 and 3 is now parallel to the ground, extending due east from the point of the pyramid. face 2 is perfectly aligned with pyramid's N face, and face 3 with the pyramid's S face. face 4 is now off the ground and doesn't align with anything in particular.

faces of the conjoined figure are now:
1: pyramid W

2: pyramid N + tetrahedron 2
3: pyramid S + tetrahedron 3
4: pyramid base
5: tetrahedron 4

with pyramid E and tetrahedron 1 glued and no longer externally visible as faces at all.

(Removed the OP-suggested image I had here. It wasn't accurate, and one commenter plus two other answers in the thread have posted much better images.)