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Math Help - Request explanation of the formula

  1. #1
    Junior Member Robocop's Avatar
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    Request explanation of the formula

    Please explain the following formulae:

    Each edge of a cube is equally divided into n parts, thus there are total n3 smaller cubes.

    1) Let N0 be the number of smaller cubes with no exposed surfaces.

    2) Let N1 be the number of smaller cubes with one exposed surfaces.

    3) Let N2 be the number of smaller cubes with two exposed surfaces.

    For (1) i.e the number of smaller cubes with no exposed surface N0 is given by the formula (n-2)3. Why is 2 subtracted from n ? And why is (n-2) raised to the power 3 ?

    For (2) i.e the number of smaller cubes with one exposed surfaces N1 is given by the formula 6(n-2)2 . Again why is 2 subtracted from n ? Why are we multiplying (n-2) with 6 ? And why is (n-2) raised to the power 2 and not 3 ?

    For (3) i.e
    the number of smaller cubes with two exposed surfaces N2 is given by the formula 12(n-2). Again why are we multiplying (n-2) with 12 ? And why are we subtracting 2 from n ?

    I will be very grateful if a detailed explanation is provided for the above formulae showing the dimensions clearly. For example Is 2 subtracted from n because it is the side layer or is there any other reason or is there any other way the cube is getting affected dimensionally? Thanks in advance!
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  2. #2
    MHF Contributor
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    Re: Request explanation of the formula

    Quote Originally Posted by Robocop View Post
    Please explain the following formulae:

    Each edge of a cube is equally divided into n parts, thus there are total n3 smaller cubes.

    1) Let N0 be the number of smaller cubes with no exposed surfaces.

    2) Let N1 be the number of smaller cubes with one exposed surfaces.

    3) Let N2 be the number of smaller cubes with two exposed surfaces.

    For (1) i.e the number of smaller cubes with no exposed surface N0 is given by the formula (n-2)3. Why is 2 subtracted from n ? And why is (n-2) raised to the power 3 ?

    For (2) i.e the number of smaller cubes with one exposed surfaces N1 is given by the formula 6(n-2)2 . Again why is 2 subtracted from n ? Why are we multiplying (n-2) with 6 ? And why is (n-2) raised to the power 2 and not 3 ?

    For (3) i.e
    the number of smaller cubes with two exposed surfaces N2 is given by the formula 12(n-2). Again why are we multiplying (n-2) with 12 ? And why are we subtracting 2 from n ?

    I will be very grateful if a detailed explanation is provided for the above formulae showing the dimensions clearly. For example Is 2 subtracted from n because it is the side layer or is there any other reason or is there any other way the cube is getting affected dimensionally? Thanks in advance!
    Just think about the geometry.

    There is a single layer of cubes on the outside that has > 0 faces showing.

    Thus the interior "cube" of cubes with faces not showing is going to be of dimensions $(n-2)(n-2)(n-2)$, i.e. n, minus the right and left layer, and minus the top and bottom layer, and minus the front and back layers. So altogether there are $(n-2)^3$ cubes w/no faces showing.



    Cubes with one face showing are on the outside layers but not along the intersection of two layers.

    There are going to be $(n-2)(n-2)$ cubes per face having a single face showing thus a total of $6(n-2)^2$ as there are six faces.



    Cubes with 2 faces showing are along the intersections of two large faces but not at the corners.

    There will be $(n-2)$ cubes along each of these intersections. There are 12 intersections. 4 at the top, 4 on the bottom, and the 4 sides so there are

    $12(n-2)$ cubes with 2 faces showing.



    It should be clear that there are always 8 cubes with 3 faces showing. These are the corner cubes.
    Thanks from HallsofIvy and Robocop
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