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Number of independent subcubes of a hypercube

  1. Mar 16, 2010 #1
    Number of "independent" subcubes of a hypercube

    Hello, I am trying to solve this problem: I have an n-dimensional hypercube and m of its vertices. Now I want to compute the maximum number of subcubes of the entire hypercube such that:
    - each subcube from the set may contain only those m vertices
    - no subcube from the set is part of another subcube from the set

    Does anybody have any idea?

    Thank you very much.

    Best regards,
  2. jcsd
  3. Apr 1, 2010 #2
    Re: Number of "independent" subcubes of a hypercube

    I take the 'subcubes' to be hypercubes of lower dimension.
    The answer depends on the positions of the m vertices critically. For each (n-1)dimensional face , find the number of vertices on it. If two adjacent faces contain 2 or more vertices, subtract the number of hypercubes on the common 'edge'. Inductively, calculate the number of hypercubes on each (n-r) D face (r=1,2,...).
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