Only the top and bottom faces of this cube are painted red; the four sides stay bare. After cutting along the lines, how many small cubes have no paint?

Only the top layer and the bottom layer of 25 cubes each touch paint. The 3 middle layers of 25 cubes are unpainted: $3\times 25=75$. (50 is the number of painted cubes; 27 would be right for a cube painted on all six faces.)