Is the Euler characteristic $\chi =2$ for the prism with a hole?

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.

$\text{Vertices} = 4\times 2\times 2=16$

$\text{Edges}=4\times2\times2+4\times 2=24$

$\text{Faces}=4\times2+2=10$

$\chi = V-E+F=16-24+10=2$.

Solution 1:

The trouble is that not all your "faces" are simply-connected.

Draw segments connecting the corner points of each square to those of the inner square. Then $V=16$, $E=32$, and $F=16$, so $\chi = 0$.

Hope this helps!