This cube is painted red on its front and back faces and blue on the other four faces, then cut along the lines. How many small cubes have only blue paint on them?

A cube with only blue paint touches a blue face but neither red face, so it lies in one of the 3 middle slices between front and back. Each slice is a 5 by 5 square whose border of 16 cubes touches blue faces: $3\times 16=48$.