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.