Math question from pre-calculus?

Question

A block is in the shpae of a rectangular prism with dimensions: n cm, (n+3) cm, and (n+9) cm, for some integer n. The surface of the block is painted and the block is then cut into 1 cm cubes by cuts parallel to the faces. If exactly half of these cubes have no paint on them, find the dimensions of the original block.

Answer 1

The total number of cubes will be:
n * (n + 3) * (n + 9) cubes.

To get the number of cubes inside, subtract 2 from each dimension:
Inside cubes = (n - 2) * (n + 1) * (n + 7)

Now expand these out...
n * (n + 3) * (n + 9)
= n (n² + 12n + 27)
= n^3 + 12n² + 27n

Inside cubes:
(n - 2) * (n + 1) * (n + 7)
= (n-2) * (n² + 8n + 7)
= n^3 + 8n² + 7n - 2n² - 16n - 14
Group like terms:
= n^3 + 6n² - 9n - 14

Remember that the inside cubes are exactly half of the total cubes, so double it:
= 2n^3 + 12n² - 18n - 28

Now equate this to the total cubes:
n^3 + 12n² + 27n = 2n^3 + 12n² - 18n - 28

Pull everything to one side (so subtract n^3 + 12n² + 27n):
n^3 - 45n - 28 = 0

Possible answers for n will be the factors of 28:
±1, ±2, ±4, ±7, ±14, ±28

By inspection, 7 works:
7^3 = 343
-45 * 7 = -315
343 - 315 - 28 = 0

So the answer is:
7 x 10 x 16 = 1120 blocks

(Note: inside you have 5 x 8 x 14 = 560 blocks)