# Thread: Need some help asap :\

1. ## Need some help asap :\

Well I'm stuck on this one math problem in my homework . I tried looking on hotmath, but sadly its a classwork problem and I have no clue how to do this or where to start So hopefully you guys can help me out

-Thanks!!!

Heres the problem:

2. Hello, scorpio!

I'll reword the problem slightly ... for convenience.

An $\displaystyle n \times n \times n$ cube is painted, then it is cut into $\displaystyle n^3$ unit cubes.

A unit cube can have: 0, 1, 2, or 3 painted faces.

Making some sketches, the table can be extended and generalized.

. . . . . . . . $\displaystyle \text{No. of painted faces}$
. . $\displaystyle \begin{array}{c||c|c|c|c|} \hline n & \quad3\quad & 2 & 1 & 0 \\ \hline \hline 2 & 8 & 0 & 0 & 0 \\ 3 & 8 & 12 & 6 & 1 \\ 4 & 8 & 24 & 24 & 8 \\ 5 & 8 & 36 & 54 & 27 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ n & 8 & 12(n-2) & 6(n-2)^2 & (n-2)^3 \\ \hline \end{array}$

Explanation

We have an $\displaystyle n\times n\times n$ cube.

There are always 8 corner-cubes (3 painted faces).

There are 12 edges; each edge has $\displaystyle (n-2)$ edge-cubes (2 painted faces).

There are 6 faces; each face has $\displaystyle (n-2)^2$ face-cubes (1 painted face).

Inside, there is a smaller cube of side $\displaystyle (n-2)\!:\;\;(n-2)^3$ unpainted cubes.