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Aug
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comment Solving geometric problems using Linear Programming
I would recommend the approach taken at CS.SE since that is $O(n)$ time versus $O(n \log n)$. I also think it lends itself to LP similar to constructing the "feasible region" (of convex shape).
Aug
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comment Solving geometric problems using Linear Programming
I believe this could point you in the right direction.
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Jul
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answered How many rectangles can fit in a polygon with n-sides?
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awarded  Autobiographer
Jul
30
comment How many rectangles can fit in a polygon with n-sides?
This question has been seen over at SO. Also, when you mention "must be consistent throughout", do you mean that all rectangles must be portrait or that their edges can only be perpendicular/parallel with each other (no weird angles)?