Quiz

We also need to consider convexity.

You can consider this a hint... Well, it seems that to create a convex figure, we cannot attach multiple parallelograms to one side. More precisely, we can, but it will be the same as one parallelogram divided in two (or three/four parts, etc.).
Why won't it be convex? Because there will be two parallel lines (two sides of the parallelograms, opposite to the sides lying on a common line) and a line intersecting them (the common side of the parallelograms). If you calculate the angles, one will definitely be greater than 180 degrees.
It sounds a bit awkward without a drawing, but I don't think anything is missing.