A great circle is a circle drawn on a sphere that is an "equator," i.e., its center is also the center of the sphere. There are $n$ great circles on a sphere, no three of which meet at any point. They divide the sphere into how many regions?
(The Art and Craft of Problem Solving, Page $37$, Problem $2.2.14$)
$2.2.12$ asks about the relation between vertices, edges and faces on a $3\text{-d}$ polyhedron, which is simply $V-E+F=2$.
$2.2.13$ asks about the number of regions an infinite plane can be split into using $n$ non-parallel lines, no three of which meet at any point. Here I checked a few cases, determined that $k=1$ and used $V-E+F=1$ to find that $f(n)=\frac 12 (n^2+n+2)$ where $f(n)$ is the number of regions.
For $2.2.14$, I am once again attempting to find $k$ such that I can use $V-E+F=k$. However, I am unsure how to define vertices or edges here. The few things I attempted all suggested different values of $k$ for different values of $n$.
I am unsure as to how I can determine $k$, as well as defining how vertices and edges are counted. Additionally, it would be really helpful if there was a simple way to manually count the number of edges, vertices and faces, especially since making $4$ great circle on a sphere and reading the number of faces, vertices and edges using Geogebra $3\text{-d}$ was very difficult.