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?