Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 exptools 1/6/84; site ihuxf.UUCP Path: utzoo!watmath!clyde!floyd!harpo!ihnp4!ihuxf!wjk 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.