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From: greg@harvard.ARPA (Greg)
Newsgroups: net.puzzle,net.math
Subject: Re: More interesting than the polar bear problem
Message-ID: <491@harvard.ARPA>
Date: Sun, 10-Nov-85 13:12:21 EST
Article-I.D.: harvard.491
Posted: Sun Nov 10 13:12:21 1985
Date-Received: Tue, 12-Nov-85 04:23:24 EST
References: <433@unido.UUCP>
Organization: Harvard
Lines: 37
Xref: watmath net.puzzle:1162 net.math:2507
Summary: Sequel to the more interesting problem.

ab@unido (Andreas Bormann) writes:
> >Johnny the adventurous flyer flies at constant altitude over the Pacific.
> >He periodically turns left by one degree.  After a while Johnny discovers
> >that his plane is at the same position *and orientation* as when he started.
> >He deduces that the path he took encloses 140 million/9*pi square
> >kilometers.
...
> After my calculations he made 352 left turns and the approximate
> air distance between two turning points was 22.237 kilometers.
> I assumed that Johnny flew an exact polygon like this:
> 
> 			    TAKEOFF
> 			  1.    |
> 			   +--==*====runway
> 			  /      \
> 		       2.+        + 352.
> 			 |        |
> 		       3.+        + 351.
> 			  \      /
> 			   +---...
> 			  4. 

This is essentially the right answer (I counted 353, but that's just a matter
of definition).  Now for an even more interesting problem:

Tom, who is Johnny's friend, is equally adventurous.  He too goes flying over
the Pacific and makes left turns by one degree.  Indeed, he also makes 353
turns and then lands on the same runway he took off of.  However, Tom does
not make these turns periodically.  He may go for hours in a straight line
and then make several turns in rapid succession.  Let X be the total area
of the Pacific that Tom's plane encircles.  What are the possible values of X?

Again, the Earth is a perfect sphere with circumference 40,000 kilometers.
Also, I should make clear that planes, when flying straight, travel in a
great circle, and do not necessarily maintain a constant compass direction.
-- 
gregregreg