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Math Help - Need some help asap :\

  1. #1
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    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:

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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, scorpio!

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


    An n \times n \times n cube is painted, then it is cut into 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.

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


    Explanation

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

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

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

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

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

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