Verified answer

mis42n's example is a surprising way to pack pyramids into a cube. But those are square pyramids (square base), not triangular pyramids.

If you call the base square of the cube A1 B1 C1 D1, and the top square A2 B2 C2 D2, with A2 above A1, B2 above B1, etc., then you can have 4 pyramids of the kind described inside the cube.

Draw the lines [A2 B1], [B1 D1], [D1 A2]. These three lines make the triangular base of a pyramid with vertex at A1. The pyramid has 3 right angles at A1 formed by edges of the cube.

The second pyramid is like the first, but with

vertex at B2, and triangle A2 B1 C2 as its base.

The third has vertex at C1 and base triangle B1 C2 D1,

and the fourth has vertex at D2 and base A2 C2 D1.

Besides these four pyramids, there is room in the cube for another pyramid with

vertex at C2 and base triangle A2 B1 D1.

This 5th pyramid is a tetrahedron, meaning that all its edges are equal, and all its triangles are equilateral triangles having all angles equal to 60 degrees.

6

I would have to say one pyramid.