Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site ucla-cs.ARPA
Path: utzoo!linus!decvax!ittatc!dcdwest!sdcsvax!sdcrdcf!ucla-cs!verma
From: verma@ucla-cs.UUCP
Newsgroups: net.puzzle,net.math
Subject: Re: Polar Bear Problem Sequel
Message-ID: <7300@ucla-cs.ARPA>
Date: Mon, 28-Oct-85 23:27:19 EST
Article-I.D.: ucla-cs.7300
Posted: Mon Oct 28 23:27:19 1985
Date-Received: Fri, 1-Nov-85 00:10:23 EST
References: <361@proper.UUCP> <855@whuxlm.UUCP> <934@turtlevax.UUCP>
Reply-To: verma@ucla-cs.UUCP (Thomas S. Verma )
Distribution: net
Organization: UCLA Computer Science Department
Lines: 88
Xref: linus net.puzzle:1024 net.math:2081

*******************__This_line_was_intentionally_left_blank__*******************

In article <934@turtlevax.UUCP> ken@turtlevax.UUCP (Ken Turkowski) writes:

{ lots of stuff we've seen 100's of times in 100's of not so unique solutions }
{ to the polar bear problem and the new polar bear problem.		      }

>
>Once you have an idea that the problem takes place on a sperical geometry,
>the answer is easy:
>
>	1/2 mile north of the south pole
>
>In a half mile you reach the pole; continue in the same direction and
>follow the rest of the steps, and you trace out a bow-tie path.
>
>However, several philosophical questions the occurs:
>
>After you reach the south pole in the first step, are you still going south?
>
>At the south pole, is there any east, west or south?
>All directions from there seem to be north.
>-- 
I don't follow you, but first lets address your second point. I think that
most people will agree with these definitions.

(underlying these is the notion of surface distance)

	Def 1: Line of longitude: A shortest path on the surface of the earth
		which connects the north pole to the south pole.

	Def 2: Line of latitude: Any line on the surface of the earth that is
		perpendicular to every line of longitude.
		(if I have these backwards, I am deeply sorry.)

	Def 3: Northern movement: Movement along a line of longitude which
		decreases ones surface distance to the south pole.

	Def 4: Southern movement: Movement along a line of longitude which
		decreases ones surface distance to the south pole.

These last two definitions lead to the following:

	Cor 1: Northern movement is just forward movement on a line of
		longitude when facing the north pole.
	Cor 2: Southern movement is just forward movement on a line of
		longitude when facing the south pole.
			

These lead to the terminology in the following definitions:

	Def 5: Western movement is just forward movement on a line of
		latitude when the north pole is to your right.

	Def 6: Eastern movement is just forward movement on a line of
		latitude when the south pole is to your right.

	Lemma 1: All of the paths which radiate outward from the south pole
		and contain no east/west movement are lines of longitude.
			

Now we can address your question (second one) more easily.

	Thm 1: Any movement from the point of the south pole is
		northern movement.

	pf: (I should do this with epsilon neighborhoods, but...)
	
	While at the south pole (a point) the only way to leave
	is to take move in a radial direction for at least some
	small distance.  Thus all paths originating at the pole
	will at least locally lie on a line of longitude.  Also
	we are leaving the south pole, therefore must be facing
	the north pole.  Hienceforth (how's that!!!) we will be 
	moving north.

That almost satisfies me, and I hope it is correct.  But back to your first
point; I do not understand your solution even if we consider moving past the
pole as continued southern movement.  First, note that since the distances
we are talking about are small (<2000 mi) compaired to the radius of the 
earth, we can assume we are in a plane.  But remember your directions though,
North is any direction out from the south pole, west is counter-clockwise, and
east clockwise around circles centered at the south pole.  So start at point
A on the circle of radius 1/2 mile.  Follow its diameter, then move counter-
clockwise for 1 mile.  We are now ~1/3 the way back (actually 1/pi).  Cross
again? and we will not be home.  I think you ment to go 1/2 circumference
of the earth >> 1 mile.  If not please re-explain.  Thank you,
					TS Verma.