A large cube of soap is dipped in blue dye so that its whole surface turns blue. It is then sliced, by cuts parallel to its faces, into 125 identical little cubes. How many of the little cubes carry dye on exactly one face?

125 little cubes means 5 along each edge. A little cube shows exactly one dyed face when it sits in the middle part of a face of the big cube, away from every edge. Each face of the big cube has a 3 by 3 middle patch, so the count is $6\times 9=54$. The traps are the other three families in the same figure: 36 is the edge count (12 edges × 3), 27 is the hidden core $3\times 3\times 3$ that gets no dye at all, and 8 is the corner count with three dyed faces.