A $4\times 4\times 4$ open cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch the bottom or one of the four lateral sides of the box?
We count the complement. A cube that does not touch the bottom or a lateral side either touches only the top side or no sides at all. These cubes form a $2\times2\times3$ prism, giving 12 cubes. These 12 cubes are subtracted from 64 to leave $\boxed{52}$.