A wildlife photographer goes one mile south, one mile east, one mile north and one mile west, and is back...
geometry/bear.square.walk New
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Problem
A wildlife photographer goes one mile south, one mile east, one mile north and one mile west, and is back at her tent; describe the bear she photographed on the way. A wildlife photographer leaves her tent one morning to take pictures. She travels exactly one mile due south but sees nothing. Then she travels exactly one mile due east, sees a bear, and takes its picture. She then travels exactly one mile due north and one mile due west, and finds herself precisely at her starting point. (Ignore vertical components of distance.) Please describe the bear. -- <U31850@uicvm.uic.edu>, January 1994 [ Editor: the first paragraph is the editor's summary. See also geometry/bear, where the walk has only three legs. ]
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[ Editor: the poser gave this partial spoiler with the problem: ] The bear is 65 cm. high at the shoulder and has brown-to-black fur with whitish-yellow markings around the eyes. -- <U31850@uicvm.uic.edu> You are effectively walking a square, but the only place where the two walks north and south won't converge is at the equator, or half a mile north of it initially. So you start off half a mile north of the equator and the two lines of longitude along which you walk north and south will be equidistant at the points at which you walk east and west thus leaving you back where you started. Sorry for not explaining that very well, but hopefully you should get what I mean. Basically if you start at any point, not half a mile north of the equator, you won't finish up where you started, a little bit west if you started north of there and a little bit east if you started south of there. The only places that I know of at the equator where you'd find bears are Borneo where you'd find the Sun or Malay bear or possibly in the Andes where you'd find the Andean bear but I'm not sure it's found that far north. But the description fits the sun bear very well I think, so I guess that's what you were looking for. -- Aidan Hollinshead <h236_010@ccvax.ucd.ie> Well, it's a complete spoiler, but to a partial answer, I think. But, as usual, there are answers near a pole. Consider the function f(x): x is distance from the south pole, f(x) is distance east of your starting point that you finish (measured in degrees). Don't bound f at 360; if you circle the pole several times, then f increases by 360 for each circuit. As you've shown, near the equator, f(x) is zero. At a distance of 1+epsilon from the south pole, f is very large. Since f is continuous, it takes on the values 360, 720, ... It's easy to see that f(5 miles) < 360, so all of those locations are near the south pole. -- Seth Breidbart <sethb@panix.com> There are also a number of solutions near the North pole. You start a distance epsilon from the pole, walk south, walk east for about 60 degrees of latitude (give or take a few degrees), walk back to your original latitude, circle the pole several times, passing through your campsite, and finally stop at the campsite. There are an infinite number of such solutions depending on how many times you circle the pole. -- David A. Karr <karr@cs.cornell.edu>