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He ends up at the North Pole.
For each meter he travels, he ends up 1/sqrt(2) meters north. Since the distance from the equator to the north pole is 10,000 kilometers, he travels a total of that times sqrt(2), or 14,142 kilometers.
3 (from a mathematical point of view): For each increment of longitude he travels, he crosses an amount of latitude equal to that increment times the cosine of his current latitude. That is:
d(lat) = d(lon) * cos(lat)
Re-arranging this and integrating, we get
d(lon)/d(lat) = 1/cos(lat)
lon = ln(tan(lat/2+pi/4))
This approaches infinity as lat approaches 90 degrees, so the traveller circumnavigates the Earth an infinite number of times.
3 (from an engineering point of view): Rearranging the final equation from (2) to solve for lat yields
lat = 2*arctan(exp(lon)) - pi/2
After passing through 360 degress of longitude, our traveller is at 89 degrees 5 minutes 13 seconds north, or 23.8 kilometers from the north pole. After 720 degrees of longitude, he's only 44.4 meters from the pole. After 1080 degrees of longitude, he's about 8 centimeters from the pole. Assuming our traveler is human, this is less than his width. Therefore, part of him is at the pole, and he circumnavigates the earth somewhat less than 3 times. (The rest of the mathematical circumnavigations is just spinning like a top.)
I enjoy the disparity (mathematically infinite, but enginneringwise less than 3) of this puzzle. |
