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From: wjk@ihuxf.UUCP (Bill Krauss)
Newsgroups: net.math
Subject: math puzzle
Message-ID: <2013@ihuxf.UUCP>
Date: Thu, 9-Feb-84 17:44:49 EST
Article-I.D.: ihuxf.2013
Posted: Thu Feb  9 17:44:49 1984
Date-Received: Fri, 10-Feb-84 07:01:47 EST
Organization: AT&T Bell Labs, Naperville, IL
Lines: 20

I've been trying unsuccessfully to solve the following problem, and I
wonder if anyone out there can help.

Given the square as shown in the picture.         |
Consider all continuous, non-self-inter-     (0,1)|_____________ (1,1)
secting curves that lie completely on or          |             |
in the square, such that one endpoint is on       |________     |
the x axis and the other is on the y axis,        |        \L   |
and neither is the point (0,0) (thus divid-       |         \   |
ing the square into two parts).  Find the         |   A      \  |
curve for which the ratio R = A / L is       (0,0)|___________|_|______
maximized, where A is the area of the part                      (1,0)
of the square that contains (0,0) and L is
the length of the curve.

I found that for the curve consisting of the segments (0,1) to (1,1)
and (1,1) to (1,0),  R = A / L = 1/2.  For the curve x**2 + y**2 = 1,
R = A / L = (pi/4) / (pi/2) = 1/2.  It seems the maximum lies somewhere
between these two examples, possibly something of the form
x**p + y**p = 1, where p > 2.