Farmer Ted Goes Natural

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.