The volume of this cube is 5*5*5 = 125 cm^3. So if you slice the cube up into smaller cubes that each have volume 1 cm^2, you end up with 125 smaller 1x1 cubes.
Each of the six faces of the cube has a surface area of 5*5 = 25 cm^2. This consists of 25 smaller squares, each with area 1 cm^2 (i.e. 1 cm by 1 cm).
Looking just at one of the faces, all of the smaller 1x1 squares on the perimeter of the larger 5x5 square (denoted by 'x' in the below diagram) will be part of a 1x1x1 cube that is either in the corner of the large cube or along one of the larger cube's edges:
x x x x x
x o o o x
x o o o x
x o o o x
x x x x x
Each of the x's denotes part of a 1x1x1 cube that will have more than one face painted, and thus we are not interested it.
Interestingly enough, notice that the o's form a 3x3 square for a total of 9. Each of these 9 is part of a 1x1x1 cube that has no other face painted (since the other faces came from inside the cube).
There are 9 such cubes gotten from each of the 6 faces. So the answer is 54.
Rich Zwelling
GMAT Instructor, Veritas Prep
