A solid cube of side 5 cm is painted red on three mutually perpendicular faces (i.e., three faces that meet at one corner) and left unpainted on the other three faces. The cube is then cut into 125 smaller cubes of side 1 cm (5 equal divisions along each edge). How many of the small cubes will have exactly one face painted?

Text-based cube explanation: Think of painting the Front, Right and Top faces of a cube, all meeting at one common corner.a Solution: Let the big cube be of size n × n × n, with n = 5. When 3 mutually perpendicular faces are painted: On each painted face, all small squares except those lying on the edges where that face meets another painted face will correspond to cubes with exactly 1 painted face. For one painted face: Total small squares = n² Squares on the 2 edges shared with other painted faces = 2n − 1 (since the common corner is counted twice) So squares on this face with only that face painted = n² − (2n − 1) = (n − 1)² For n = 5: (5 − 1)² = 4² = 16 small cubes with exactly 1 painted face on each painted face. There are 3 such painted faces, and these sets do not overlap (we have excluded intersections). Total cubes with exactly 1 face painted = 3 × 16 = 48.