I am interested in knowing what the most/least common patterns (not necessarily stable) that can occur inside a NxN square. I know that there was some census of 3x3 patterns done with relation to the backtracking searcher a while back, but I'm not sure that exact issue was addressed. I believe that for all grid sizes, the most common pattern is empty space, but the question of the least common pattern is more difficult to answer. I think that for 2x2, the solution is
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For 3x3, there are a number of options that I would consider:
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x = 63, y = 3, rule = B3/S23
3o2b3o2b3o2bobo2bobo2b2o3bobo2bobo2bo4b2o3bo4bo5bo$3o2b3o2b3o2b2o3b2o
4bo4bo3b3o2b3o12bobo3b2o$3o3b2o2bobo2b3o3b2o2bobo2bobo2bobo2bobo2bobo
2bobo2bobo2b3o!
I think 4x4 is a much more difficult question to answer. There are a multitude of options to choose from (not just these):
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x = 53, y = 4, rule = B3/S23
2obo3b4o2b2obo2bob2o4bo3b2o4bo2bo2bobo3b2obo$obo4b2obo3bobo2b2obo2b2o
6b2o2b4o3b3o3bo$bobo3bob2o2bobo3bob2o4b2o2b2o4b4o2b4o2bob2o$ob2o3b4o2b
ob2o2b2obo3bo6b2o2bo2bo3b3o3bobo!
This problem cannot continue indefinitely to larger grid sizes; GoE patterns would have a frequency of exactly zero.
