You have a rectangle of dimensions $l \times b$ where $l\geq b \geq 1$ and $l,b \in \mathbb R$. How many circles of diameter one unit can be fitted inside this rectangle without overlapping? (Touching of circles is allowed)
I thought this was a simple problem and googled it. And while wikipedia mentions that a hexagonal packing arrangement is best, sometimes packing more circles along the length is better, sometimes packing more circles along the breadth is better. And I'm not convinced that one of the above two hexagonal arrangements is always optimal.
So is there a simple solution for large numbers of length and breadth?
Also a reference table or tip while tackling this problem for not-so-large length and breadth will also be helpful.