Are there Monohedra with odd number of faces? A monohedron is a convex polyhedron with all faces mutually congruent but with no other symmetry necessarily needed. So obviously, this is a wide class of polyhedrons that includes the Platonic solids and isohedra (https://mathworld.wolfram.com/Isohedron.html). An earlier post is https://math.stackexchange.com/questions/3888486/what-are-the-known-convex-polyhedra-with-congruent-faces
Questions: Are there monohedrons with odd number of faces (it is known that isohedrons necessarily have even number of faces - as stated in https://mathworld.wolfram.com/Isohedron.html)? What are the values for the number of edges on a face for which monohedrons are possible? Will relaxing convexity (of the body, not of the faces) have an impact on the answers to these questions?
 A: This is too long for a comment.
The figure shows how to construct an $11$-face convex polyhedron whose faces all quadrilaterals, but not congruent. I think it's only interesting because it shows that there aren't plain combinatorial or topological obstructions to a positive answer to the question. On the other hand I doubt it's possible to deform this specific construction to produce a monohedron (I don't see how to make all the faces have the same number and order of obtuse and acute angles, never mind corresponding edges of equal length).
Start with a hexagonal prism. Cut the top hexagon in $2$ and the bottom one in $3$, to created $5$ degenerate (coplanar) quadrangular faces in place of the two hexagons. Slide G, L and M out a bit to make the faces not coplanar, but since this may destroy the flatness, slide H, I, J, K, C, E, B in a bit, each as necessary, to make all the faces flat again.

A: The answer to the question in the title is negative in dimension 3: it was shown by Grunbaum that every 3-polytope with congruent facets has an even number of facets. See p. 414 of his book "Convex Polytopes", 2nd edition. The original reference is:
Grunbaum, B. On polyhedra in $\mathbb{E}^3$ having all faces congruent. Bull. Res. Council Israel, 8F (1960), 215-218.
