THE FOUNDATION

Dr. Greg Martin, a friend of mine from graduate school who is now at the University of British Columbia, published the 1999 paper "Farmer Ted Goes Natural" in The Mathematics Magazine, v. 72, n. 4, pp. 259-276. In that paper, he defines the concept of "almost-square." He characterizes these integers, studies their asymptotic behavior, their computatibility, and other interesting results. What is an almost-square, you ask?

In Martin's paper, he reminds us of the following problem from calculus class. "Farmer Ted is building a chicken coop. He decides he can spare 190 square feet of his land for the coop, which will be built in the shape of a rectangle. Being a practical man, Farmer Ted wants to spend as little as possible on the chicken wire for the fence. What dimensions should he make the chicken coop?"

Any good calculus student knows that the chicken coop should be a square with side lengths 1901/2 feet. Unfortunately, if Poor Ted were to go to a hardware store and ask for exactly these dimensions, he would probably be laughed at. Hence, perhaps he would do better to try to determine what chicken coop of integer side lengths and area less than or equal to 190 feet would be the most cost effective in terms of its area to perimeter ratio.

For a positive integer n, let s(n) denote the least possible semiperimeter of a rectangle with integer side lengths and area n. Let F(n)=n/s(n). Then Martin defines an almost-square to be a natural number n such that F(k)<=F(n) for all k<=n.

In his paper, Martin suggested several possible avenues of further research, two of which are explained below.

SUMMER 1999

Shawn Alspaugh, then finishing his Freshman year at Taylor, did a summer research project with me generalizing some of Martin's results. Alspaugh took the leap up a dimension, and defined almost-cubes. These are defined analogously to almost-squares. (In fact, there are two plausible ways to define an almost-cube. Alspaugh looked at those right rectangular parallelipipeds with high volume to surface area ratio, rather than those with high volume to sum-of-side-length ratio.)

Alspaugh, now a graduate student at Indiana University, proved several interesting lemmas relating almost-cubes to almost-squares. He also gave a complete characterization of almost-cubes, although it is not as computationally friendly as Martin's characterization of almost-squares. Alspaugh's work is detailed in his paper "Farmer Ted Goes 3D," which has been accepted for publication in The Mathematics Magazine. The sequence of almost-cubes can also be found in the On-Line Encyclopedia of Integer Sequences by entering "1,2,3,4,6,8,12,16,18,24,27,32,36,45,48,54,60" at the prompt.

SUMMER 2001

Bobbe Cooper, then finishing her Freshman year at Taylor, did a summer research project with me also generalizing some of Martin's results. Cooper looked at alpha-almost-squares, which arise from generalizing Martin's problem in the way familiar from many calculus text books, that is by specifying that one side of fencing is made of more expensive material than the others. In other words, consider the cost-perimeter of a rectangle of area A=xy, which would be x+alpha*y. Alpha-almost-squares are then defined analogously to almost squares, with minimim semiperimeter replaced by minimum semi-cost-perimeter.

Cooper, who has been accepted to the graduate program at the University of Georgia, completed a characterization of alpha-almost-squares for any integer alpha greater than or equal to one. Her work is detailed in her paper "A Characterization of Alpha-Almost-Squares," which is also slated to appear in The Mathematics Magazine after suitable revision. A list of some Alpha-Almost-Squares and the Perl script used to generate them can be found here.

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