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Math Help - Arranging cross shaped particles on a 2D grid

  1. #1
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    Post Arranging cross shaped particles on a 2D grid

    Suppose we have a 2D grid. This grid has L sites in the x direction, and D sites in the y direction. Further suppose that the y dimension has periodic boundaries, and the x direction is bounded by hard walls.

    In this L by D grid we place N cross shaped particles. They are arranged so that every particle occupies 5 grid sites as shown below.

    o o o o o
    o o x o o
    o x x x o
    o o x o o
    o o o o o

    The crosses are mutually repulsive, so if you place a cross center in a position (depicted below by C), you cannot place another in the grid points depicted by x:

    o o x o o
    o x x x o
    x x C x x
    o x x x o
    o o x o o

    It is also impossible to place cross centers at the most extreme grid points in the x direction, as one of their points would be out of the grid.

    Suppose that the grid is large enough to accomondate N crosses under aforementioned restrictions.

    The question is as follows:
    With L, D and N known, and freely distributing the crosses, what are the possible values (and possibly probabilities) of the occupation of the last column of L.
    Rephrasing: how many grid points on the last column of the grid (on the x direction) are occupied? of course there may be a lot of options, but there is a strict range of possible ocupation numbers (and probability associated with each one).

    For example:
    L = 3 ; N = n ; D >> N
    There's only one possibe occupation numer, which is N.

    Another example:
    L = 4 ; D = 3 ; N = 1
    There are two options, either 0 or 1, with equal probabilities.

    L = 5 ; D = 3 ; N = 1
    Same two options, 0 or 1, but now with probabilities 2/3 and 1/3 respectively.


    Anyone has an idea of how to attack this problem for general L, D, N?
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  2. #2
    Super Member malaygoel's Avatar
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    You said that y-dimension has periodic boundaries and x-direction is bounded by hard walls.........does this has anything to do with the question?
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  3. #3
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    It does. It means that a cross center cannot be placed in points such as (0,y) or (L,y) in the x direction. In the y direction, cross centers can be placed in (x,0) or (x,D), but then the excess part of the cross protrudes from the other side and denies other crosses to be placed in its nearest neighbors.
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