Path On The Surface Of The Earth is a neat one, took a while to realize why there were multiple answers. SPOILER: Obviously the North Pole is one answer, becau

Path On The Surface Of The Earth is a neat one, took a while to realize why there were multiple answers. SPOILER: Obviously the North Pole is one answer, because if you're 1 km south of the pole then it doesn't matter what longitude you're at, if you go 1 km north you'll be back at the pole. The other answers are more interesting, though, and involve the south pole. The key observation is that at certain locations fairly close to the south pole, going east by 1 km will bring you back exactly where you started because the longitude circles are very small, so there's a set of latitudes where your 1 km journey will wrap you around the world any number of times. From that observation you can figure out the rest if you care to flex your trig...