How to prove the number of images of two mirrors inclined at $A$ is $360/A -1$ Consider two mirrors constructed as follows.

How can we prove the number of images  is $360/A -1$? How about the case in which $360/A$ is not an integer? Seriously I have no idea how to start solving it analytically rather than by using geometric construction with compass and straightedge.
 A: 
How can we prove the number of images is $360/A−1$?

Have you tried drawing this? I.e. draw both the original slice and all its images in the same plane. That should give you a clear idea as to where this formula comes from.

How about the case in which $360/A$ is not an integer?

In fact, $\frac{180°}{A}\not\in\mathbb N$ will be a problem, even if $\frac{360°}{A}\in\mathbb N$, since in that cases you get an even number of mirror images, which together with the original make an odd number of objects. This doesn't fit in with the orientation changing between an image and its mirror.
For all these cases, you will notice a seam: the image you see to the left of the common edge between the two mirrors (if you think about mirrors in 3D; it would be the single point of intersection in the plane) will not fit in with the image to the right.

Seriously I have no idea how to start solving it analytically rather than by using geometric construction with compass and straightedge.

Who says you have to do this analytically?
One way to proove this analytically would be expressing the two reflections as operations of some reasonable kind (e.g. matrix multiplication, or a combination of complex multiplication and complex conjugation), and then have a look at the group generated by these operations. You will find that the resulting group is finite iff $\frac{360°}{A}\in\mathbb N$. You will also notice that for $\frac{180°}A\not\in\mathbb N$ there exists a group element which maps the slice between the mirrors onto itself but is not the identity but instead a reflection along the middle of that slice. Which leads you to $\frac{180°}A\in\mathbb N$ as the neccessary and sufficient condition for a finite number of images which do not overlap one another.
