Is the Euler characteristic $\chi =2$ for the prism with a hole?
In attempting to lead a bunch of high school students to an understanding of the Gauss-Bonnet theorem, I've run into a problem determining the Euler characteristic of a cube with a square hole through it. The problem is that the surface is like that of a torus, and so ought to have $\chi = 0$, but a simple counting of $V$, $F$, and $E$ yields $\chi = 2$.
I have a wide range of abilities in my class, so I need to motivate this theorem without the use of calculus. My hope was that by examining polyhedra of different types, they would come to "discover" a connection between the Euler characteristic, the total angle defect, and the number of holes through the surface, but now I'm stuck trying to come up with a way to rig the counting to make $\chi$ come out right.
I would love some insight on
- how to make the count come out to $\chi = 0$
- how to justify such a counting method so that it does not seem arbitrary to my students.