The top and bottom of this cube are red, and the four sides blue. After it is cut along the lines, how many small cubes show blue but no red?

Cubes with no red lie in the 4 middle layers. In each such layer of $6\times 6$, the cubes on the outer ring touch a blue side: $36-16=20$. So $4\times 20=80$. (96 counts all blue squares on the four side faces.)