Geometry/Geography Riddle
Posted by Rob Herman at August 31st, 2006
I’m moving, so riddles will be the order of the next couple of updates. That’s my excuse and I’m sticking to it.
You start at a particular spot on the Earth. You walk a mile south, then a mile east, then a mile north. You end up exactly where you started. Desctibe the set of all points where you could have started. Assume the Earth is a sphere.
Hint: The number of points is not 1.
I was actually asked this at an interview, and my failure to work it out correctly at the time was part of the reason I didn’t get a second one. (Have figured it out since then)
If the Earth is assumed to be a perfect sphere, wouldn’t all possible starting points be equal? Woudn’t, then, the answer be narrowed down to either no points or all points?
Gary: Consider the difference between traveling north south and traveling east west on a sphere. You can think of this difference as the difference between longitude and latitude. For example there are two points (the poles) where it is impossible to travel east or west. At all other points however it is possible to travel east or west.
Part of the Solution: Obviously the north pole is one point for this riddle. Walk a mile south putting you on latitude X, walk a mile east (still on latitude X) then walk a mile North putting you once more at the north pole.
The other part: Any point a mile north of a latitude that has “length” that evenly divides 1 mile. Where the length of a latitude is difined as the distance you would cover while walking on that latitude (either east or west) when you get to the same point you started at.
My sixth grad teacher capped it off with “What color was the bear that ate him?”
Gary: The sphere still rotates about an axis. The north and south poles are the ends of the axis and east and west have their usual meanings WRT north and south.
DrObviousSo: I don’t like that addition to the puzzle because it only really acknowledges the more obvious answer; there are no bears in Antarctica, which is where infinitely more of the solutions lie…
Unfortunately, with global warming, it’s no longer reasonable to assume you can make a walking trip starting at the North Pole! It’ll have to be changed to “walk or swim a mile…”
The following latitudes are solutions to the riddle
90 deg 0 min 0 sec North (north pole)
89 deg 58 min 59.671 sec south
89 deg 59 min 3.812 sec south
89 deg 59 min 5.193 sec south
89 deg 59 min 5.884 sec south
89 deg 59 min 6.298 sec south
89 deg 59 min 6.574 sec south
89 deg 59 min 6.771 sec south
89 deg 59 min 6.919 sec south
89 deg 59 min 7.034 sec south
89 deg 59 min 7.126 sec south
etc …
There is an infinite number of latitudes farther south that fit the description however the last two listed here are only ~ 9 feet 4 inches apart. The calculation incase any one is curious is
[inverse cosine](1/(2N[Pi]X)) - 360/(2X[Pi]) Where N is an integer > 0 and X is the radius of the world in miles ( ~ 3963.1675996952 according to google)
Also if you consider standing at the south pole and spining in circles while freezing to death “traveling east” then 89 deg 59 min 7.94 sec south is the southern most solution since that is 1 mile north of the south pole.
so basically mt. everest is also a solution because if you start walking south and and then east, which is the circumference, and then north, you end up at the exact place you were.
Anonymous: Not quite, because travelling east won’t take you around the circumference; you’ll start going around briefly, but will then find yourself walking off the mountain or having to curve north to stay on its circumference.