Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83 (MC840302); site unido.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!philabs!cmcl2!seismo!mcvax!unido!ab
From: ab@unido.UUCP (Andreas Bormann)
Newsgroups: net.puzzle,net.math
Subject: Re: More interesting than the polar bear problem
Message-ID: <433@unido.UUCP>
Date: Sat, 9-Nov-85 07:58:37 EST
Article-I.D.: unido.433
Posted: Sat Nov  9 07:58:37 1985
Date-Received: Mon, 11-Nov-85 06:59:49 EST
Organization: University of Dortmund
Lines: 59
Xref: watmath net.puzzle:1161 net.math:2505

>/***** unido:net.puzzle / harvard!greg /  5:41 am  Nov  4, 1985*/
>Subject: More interesting than the polar bear problem
>
>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.
>Therefore he lands. He deduces that the path he took encloses 140 million/9*pi
>square kilometers.  Now for some questions:
>
>1)  How many left turns did Johnny make?
>2)  What was the approximate air distance between two consecutive turns?
>
>You may assume that the Earth is a perfect sphere with a circumference of
>exactly 40,000 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.      

But when he reached his starting point he had to make one more 1deg-turn
to get into the same orientation as the runway. So my solution might
not be correct. Maybe there are an infinite set of solutions looking
this way:
			  TAKEOFF last turning point
			  1.    | |
			   +--==*=+==runway
			  /        \
		       2.+          + n-1.
			 |          |
		       3.+          + n-2.
			  \        /
			   +---   
			  4.      

Possibly the number of left turns is 353 in any case...


	Andreas Bormann
	University of Dortmund  [UniDo]
	West Germany

Uucp:   ab@unido.uucp
Path:   {USA}!ihnp4!seismo!mcvax!unido!ab
	{Europe}!{cernvax,diku,enea,ircam,mcvax,prlb2,tuvie,ukc}!unido!ab
Bitnet: ab@ddoinf6.bitnet
		 \
Missiles:    -=>-->-*>      N 51 29' 05"   E 07 24' 42"
		 /