A cube is painted red on its front and back faces and blue on the other four faces, and then cut along the lines into 64 equal small cubes. How many small cubes show both colours?

Both colours appear only in the front and back layers, on their outer rings where a red face meets a blue one. Each ring has $4\times 4-2\times 2=12$ cubes, so $2\times 12=24$. (16 counts only the edge cubes and misses the 8 corners; 32 is all of the two red layers.)