I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids.
Help me find my error. I'd appreciate if someone could just point out which of V, E, F is wrong.
$\text{Vertices} = 4\times 2\times 2=16$
$\text{Edges}=4\times2\times2+4\times 2=24$
$\text{Faces}=4\times2+2=10$
$\chi = V-E+F=16-24+10=2$.