TheGerman wrote:
The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?
A. 6n^2
B. 6n^2 – 12n + 8
C. 6n^2 – 16n + 24
D. 4n^2
E. 24n – 24
problem never specifies a real number for any of the steps, so select a smart number for n, say n = 3. In this case, the original cube is painted and then cut into a 3 × 3 × 3 assemblage of 27 smaller cubes. Imagine the top face of the original (large) cube (if you know what a Rubik’s Cube is, picture one!). Every cube on that face has been painted on at least one side. The same is true for the bottom face. Now think about the middle “slice” of the cube. This “slice” contains a total of 9 cubes, but only the one in the very middle has not been painted on any of its sides.
Therefore, only one of those 27 cubes—the one in the middle of the structure—remains unpainted. When n = 3, the answer is 26.
Plug n = 3 into the answers and look for 26:
(A) 6(32) = 6(9) = 54
(B) 6(32) – 12(3) + 8 = 54 – 36 + 8 = 26
(C) 6(32) – 16(3) + 24 = 54 – 48 + 24 = 30
(D) 4(32) = 4(9) = 36
(E) 24(3) – 24 = 72 – 24 = 48