A square shaft two cubes wide is cut straight down through the middle of a $4\times 4\times 4$ block of small cubes. The block is then painted everywhere, including inside the shaft. How many small cubes have exactly three painted faces?

Each layer is a ring of 12 cubes. In the top and bottom layers, every cube has the top (or bottom) painted, plus either two outer sides (the 4 corners) or one outer side and one shaft wall (the other 8): three faces each, 24 cubes. In the two middle layers each cube has only two painted faces. So 24.