Problem Solving

josh80 wrote:The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n³ 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) 6n²
B) 6n² - 12n + 8
C) 6n² - 16n + 24
D) 4n²
E) 24n - 24

Hey josh80,

When posting questions, be careful with your exponents. Writing n3 can be confusing.
To show exponents, you can use "^" as in "...the cube is sliced into n^3 smaller cubes"


Okay, a fast approach here is to examine a specific case (i.e., a specific value of n) and compare the result to the answer choices.

So, let's take a wooden cube and slice it into 3³ smaller cubes (i.e., n = 3).


There are 27 smaller cubes altogether, and ONLY 1 of them (the small cube in the very center) does not have paint on it. So, there are 26 cubes that have paint on them.

So, when n = 3, there are 26 cubes that have paint on them.
Now, we'll check the answer choices and see which one yields a value of 26 when n = 3

A) 6(3)² = 54 NOPE
B) 6(3)² - 12(3) + 8 = 26 PERFECT!
C) 6(3)² - 16(3) + 24 = 30 NOPE
D) 4(3)² = 36 NOPE
E) 24(3) - 24 = 48 NOPE

Answer: B

Cheers,
Brent