back to the same spot

 In the post immediately prior to this I talked about a Youtube short in which people were asked how long someone can serve as US President.

Today I saw a similar short in which someone was asked the following riddle:

You are standing on the earth. You walk one mile south, one mile west and then one mile north. Where are you.

In the video, it's presented as something Elon Musk asks all potential employees. I actually doubt that that part is true, but whatevs. In the video I saw, the answer given was the North Pole. Because, if you start at the North Pole, go south for a mile and then west for a mile, you are still one mile south of the North Pole. So when you finish by going north, you wind up at the North Pole again.

That answer is good as far as it goes. But, in fact, it's not the only answer.

Near the South Pole, there is a line of latitude at which the distance around the world is exactly one mile. If the earth were flat, then using the rules of geometry we could solve for the latitude and find that it's 1/(2×pi) miles (about 840 feet) north of the South Pole. That's actually off by a little, since the earth is a sphere. But at such scales, that distortion is tiny. At any rate, there is a circle around the South Pole that's exactly one mile around. Any point exactly one mile north of that circle is a correct answer to the riddle. Because if you start one mile north of the circle and go one mile south, you will be on the circle. Travelling one mile west puts you back at the same spot on the circle, and one mile north puts you back where you started. So there's a whole circle of correct answers.

But it get's even better!

At about 420 feet north of the South Pole, there's a circle that's exactly half a mile around. Any point one mile north of that circle is a correct answer.

And at about 280 feet north of the South Pole there's a circle that's exactly a third of a mile around. And at about 210 feet north of the South Pole there's a circle that's exactly a fourth of a mile around. Any point exactly one mile north of either of these circles is a correct answer. For any positive integer N, there's a circle around the south pole that's exactly N miles around. Any point exactly one mile north of any of these circles is a correct answer.

So take that, Elon.