Take the 2-minute tour ×  Imagine I generate some polyomino
(http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares,
under the constraint that I want to maximize the number of
shared edges between unit squares, or equivalently, minimize the
effective surface area of the tiling.  To be a bit more
specific, say I give you $N$ unit squares and ask you to do as
you wish with them in terms of making a connected polyomino.
Once you've finished, I then count how many many of the $4N$
tile edges are shared between two tiles. Your goal is to
maximize this count.  How well can we do as a function of $N$?
What tiling geometry is optimal - perhaps a rectangle or a
square?  We are inspired by this earlier question: (Polyominoes
with double contact)  share|improve this question Can I pick
them or are they given to me? Are they congruent? In general it
can be quite hard to decide if a set of polyominos (even if they
are congruent) tile a rectangle. By surface area do you mean the
smallest rectangle containing all the