The “Minimum Vertices” periscope as of the end of [22. Merging Vertices in Blender] is more complex than it needs to be.
As a first warning sign, take Euler’s Formula. We know that for any convex shape, Faces + Verticies - Edges should equal 2. For the model as of the end of the lecture, we have [14] + [12] - [22] = 4. Some somewhere, we have some extraneous information. It happens to be that the model at the end of the lecture has two internal faces.
Additionally, even more simplification is possible. I was able to get the basic periscope shape down to [Verts:8 | Edges:12 | Faces:6]. This is simply each of the shape’s 8 vertices joined together by edges. This is the simplest form that the shape can be in, and thus the cheapest this mesh can be.
