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    Glad to know I had a positive influence on your life.
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    This is a great classic whose complete solution is more intricate than it seems at first glance. The intended solution of the puzzle was probably "at the North Pole" (see figure to the right). This first solution is so obvious and overwhelming that it is tempting to stop at that and overlook a whole set of totally different solutions near the South Pole:
    The second leg of your journey ("due east") could be a circular path one kilometer in circumference around the South Pole (its radius is about 159.155 m). Therefore, "home" could be anywhere at a distance of about 1159.155 m from the South Pole: The first leg of your journey (due south) will lead you to this circular path at some point A. Walking due east for 1000 m will take you all the way around the circle back to point A from which you will get home by walking a kilometer due north...
    Now, it's also possible to have a circle 500 m in circumference as the second leg of the trip. By walking 1000 m on it, you just travel twice around the circle but end up back to A just the same. This means that "home" could also be about 1079.577 from the South pole. Obviously now, for any positive integer n, any circular path 1/n km in length would be an acceptable second leg of your walk. All told, there is an infinite family of circles around the South Pole were your "home" could be.
    Final answer: Your home is either at the North Pole or at a distance from the South Pole roughly equal to (1 + 1/2pn) km , for some positive integer n.
    NOTE: I had to say "roughly equal to" because the circumference of a small "circle" drawn on a curved surface is not quite equal to 2p times the "radius" which can be measured on that surface:
    On a sphere of radius R, the distance d measured on the surface is along an arc of a great circle (that's indeed the shortest possible distance). If the "radius" d of a circle is measured in that way, its circumference is only 2p[R sin(d/R)] (since the bracketed quantity is the actual radius of that circle in 3D space). For small values of x=d/R, we have sin(x) » x (1-x 2/6), and 1/sin(x) » 1/x (1+x 2/6).
    Before we apply this to the above situation, we may stop to consider what is theoretically the best value to use for R around the South Pole (this may seem like a ludicrous concern --and it is-- but we are already into ludicrous precision at this point, so we may as well learn something from this): First, consider the Reference Ellipsoid (the defined regular shape with respect to which professionals are charting the irregularities of the Earth's so-called "sea-level", which is an equipotential surface averaged over time at each point of the Globe). Its "equatorial radius" is defined as precisely equal to a=6378137 m, whereas the "polar radius" (i.e. half the distance between the North Pole and the South Pole) is about 6356752.3141404 m. This makes the radius of curvature of the Meridian at the South Pole equal to a 2/b or 6399593.6258639 m, for this surface of reference. Now, the South Pole is not at sea level, but on an elevated plateau at about 2835 m of altitude (with an unbelievably thick layer of about 8800' feet of ice, which accounts for almost 95% of that altitude). This altitude is to be essentially added to the radius of curvature of the "sea-level" reference meridian. Therefore, if the surface at the South Pole was perfectly level and smooth, it would be extremely close to the surface of a sphere with a radius R of about 6402428 m. This would make the tiny quantity 1/(6 R 2 ) (used in the "final" result below) equal to about 4.066 10-9
    All told, we may now state with more (ludicrous) precision that "home" could be either on the North Pole or at a distance from the South Pole very close to:
    [ 1 + 1/2pn (1 + (4.066 10-9)/n 2 ) ] km , for some positive integer n.
    One last word about points at a distance of exactly 1 km from the North Pole: If your "home" is there, walking a kilometer south will bring you to the South Pole itself. From there, you can only walk north and could therefore not perform the second leg of the trip ("due east") at all. Such points are therefore not solutions of the problem as stated (but there are infinitely many solutions in any neighborhood of them which are solutions. Of course no point whose distance to the South Pole is less than 1 km; can be a solution either, since such a starting point does not even allow you to complete the first leg (i.e. walk 1 km due south)...
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Where in order to get to the South, you have to go north


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