  A traditional "Farmer Ted" calculus problem is to minimize the perimeter of a
rectangular chicken coop given the area N, so that as little as possible will
be spent on the fencing. But what if N is an integer, and we are only allowed
to consider rectangles with integer side lengths? Often it will be more
cost-effective to build a coop with area smaller than N, where the measure of
cost-effectiveness is the ratio of the area to the perimeter. Those numbers N
that are the areas of rectangles that are more cost-effective than any smaller
rectangle are dubbed "almost-squares", in deference to our intuition that such
numbers ought to be the product of two nearly equal factors. This paper
investigates the characterization and distribution of the almost-squares. It is
shown that almost-squares can be equivalently described in a surprisingly
elegant way, and that computing whether a number is an almost-square and
computing the least almost-square not exceeding N can be done surprisingly
efficiently (much faster than factoring integers, for instance). Several bad
jokes are included.
