No, of course not. I mean, the approximation is going to be somewhat close, right? 1/4 is .25, whereas 1/2pi is .16
On a sphere (which the Earth is not–it bulges 20 km farther at the equator than the pole, but we’ll ignore that) of radius R, at a latitude (we can assume that the geodetic latitude is the same as geocentric latitude since we are assuming the Earth is a sphere) L south, the radius of the circle formed by following a constant easterly direction is R times the cosine of L. The circumference is 2pi x R x cos(L)–and you want that to be an even fraction 1/N of 1 mile. In other words, L is the arccos(1/(2pi x R x N)). Mean radius of the Earth is about 6371 kilometers or 3959 miles–which we can use for our purposes.
The distance from the pole to that circle is (90-L) x pi/180 x R, where the pi/180 is necessary to convert degrees into radians. Then, you’d add a mile to get the starting point.
I calculate that starting point (for N equal one–that is, you’d only go around the circle once) to be 1.1592 miles from the South Pole, very close to DrMatrix’s answer. As you’d expect, since the Earth is so large the curvature is not going to make much of a difference. In fact, if we hadn’t rounded off our answers, his answer and mine would agree for ten decimal places.
Thanks for the replies (well, the ones not about bears :)).
I realized when I woke up this morning that r is only equal to the height from the pole(s) at the equator (assuming a sphere) and that within a mile or so from the pole the curve is so small you might as well assume it was flat (if you forgot, as I did, the equation so kindly supplied by RM Mentock which comes to an answer very close to the flat approximation (as DrMatrix said).
Thanks for pointing out my ‘loose’ spelling yanx4ever, but why would you not be able to start 1.159 miles north of the south pole and still end up where you started (baring impassable mountains and such).
Indeed you can, PosterChild; yanx4ever just needs to think about it more. I’d just like to say that I really like your technique for hiding the answer to the puzzle. If you came up with this yourself, then you get a bid “good show!”
Thanks. I learned that trick by clicking “quote” on posts that did cool things, but I’m glad to know (thanks again to RM Mentock!) the link to the list of vb code.
I stand corrected- But I think there are many answers to the puzzle.
Obviously if you start at exactly 90 degrees N latitude, if you go 1 mile south, any distance or none east or west, and 1 mile north you wind up where you started.
You are correct, if you start out 1.1592 miles away from the South Pole, when you walk 1 mile south you are 0.1592 from the pole, then going east one mile would make you go “around the world” once, then 1 mile north to your starting point.
But suppose…
You start out 1.0796 miles from the South Pole. Going 1 mile south puts you 0.0796 miles from the pole, in which case 1 mile east makes you go “around the world” exactly twice, thence 1 mile north to your starting point… which leads to
You start out 1.0531 miles from the South Pole. Going 1 mile south puts you 0.0531 miles from the pole, in which case 1 mile east makes you go “around the world” exactly three times, thence 1 mile north to your starting point… so let’s take this to a somewhat logical conclusion…
You start out 1 mile plus 0.4775 feet from the South Pole. Going 1 mile south puts you 0.4775 feet from the pole. At this point if the actual pole that marks the spot has been recently spotted correctly, you put the right side of your body against it, and keeping your feet exactly 0.4775 feet from the pole, you run around it exactly 1760 times, making you go east exactly 1 mile, thence north 1 mile to your origin.
You are all absolutely right- the original DrMatrix answer was the definitive answer. Although there are practical limitations on the size of N- as N grows too large the circle around the pole becomes too small to physically walk, particularly when chasing or being chased by a bear (even if it doesn’t belong there).
What was when I heard it years ago as a kid riddle has more layers than first comes to mind- thanks to all of you sharpies that opened up my eyes.