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From: mcewan@uiucdcs.CS.UIUC.EDU
Newsgroups: net.math
Subject: Re: Polar Bear Problem Sequel
Message-ID: <28200056@uiucdcs>
Date: Thu, 24-Oct-85 00:44:00 EDT
Article-I.D.: uiucdcs.28200056
Posted: Thu Oct 24 00:44:00 1985
Date-Received: Fri, 25-Oct-85 04:03:11 EDT
References: <855@whuxlm.UUCP>
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Nf-ID: #R:whuxlm.UUCP:-85500:uiucdcs:28200056:000:923
Nf-From: uiucdcs.CS.UIUC.EDU!mcewan    Oct 23 23:44:00 1985


>
>	Where on the earth can one walk 1 mile south, 1 mile west, 1 mile
>	north, AND 1 mile east, and end up at the starting point?
>
First and most obviously, anywhere on the 2 circles 1/2 mile north and south
of the equator.

Next, on any circle near the south pole satisfying the following conditions:
distance from the pole is 1+x miles, where x satisfies:

	2*pi*n*x - x/(1+x) = 1,	n = 1,2, ...

From here, walking 1 mile south lands us on a circle x miles from the pole.
Walking 1 mile west, we end up x/(1+x) miles east of our last stop.
One mile north of here is one mile east of our starting point.

Solving for x:

	x = [-n*pi + 1 + sqrt(n^2*pi^2 + 1)]/2*n*pi,	n = 1, 2, ...

Similarly, any point on a circle x miles from the north pole will do.
I think that's it.



			Scott McEwan
			{ihnp4,pur-ee}!uiucdcs!mcewan

"There are good guys and there are bad guys. The job of the good guys is to
 kill the bad guys."