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Math Help - Slicing a parallelopipe

  1. #1
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    Slicing a parallelopipe

    Hi, if there is a rectangular parallelopipe x*y*z ,made of 1*1*1 units then into how many parts can it be sliced by appling cuts at most n times parallel to it's parallel sides?
    E.g if x=y=z=2 and n=3
    then number of parts = 8.
    If x=y=z then i think answer is x^n
    but what for other cases?
    Note: the cuts are made along integer co-ordinates.
    Thanks.
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  2. #2
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    Quote Originally Posted by pranay View Post
    Hi, if there is a rectangular parallelopipe x*y*z ,made of 1*1*1 units then into how many parts can it be sliced by appling cuts at most n times parallel to it's parallel sides?
    E.g if x=y=z=2 and n=3
    then number of parts = 8.
    If x=y=z then i think answer is x^n
    but what for other cases?
    Note: the cuts are made along integer co-ordinates.
    Thanks.
    I'm not sure that I understand your question completely and correctly. So I'm guessing:

    1. Let N_{xy}, N_{xz}, N_{yz} denote the number of cuts parallel to the xy-plane, xz-plane and yz-plane respectively. With each cut you'll get N +1 pieces.

    Then the number A of pieces which you'll get by slicing the parallelepiped is calculated by:

    A = (N_{xy}+1) \cdot  (N_{xz}+1) \cdot (N_{yz}+1)

    2. If N_{xy} = N_{xz} = N_{yz} then A = (N+1)^3
    Last edited by earboth; April 10th 2011 at 07:56 AM.
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