Your time travelling has caused a few issues with the space-time continuum, like your old high school bully, Riff, now ruling the future with an iron fist. You and the professor have driven your hovering DeLorean back to the past to undo your own meddling. But shortly after you fix your mistakes, another DeLorean appears and crashes into yours. Out step older versions of you and the professor, who apparently had the same idea from somewhere else in the timeline. The professors panic and explain that the universe could collapse now that you’re both in the same time and place. The only hope is to merge the time streams by having each version of you simultaneously time travel home. You both have plenty of fuel, but the crash broke your chrono-stabilizing gyroscopes and your cars can only time travel under very specific circumstances. Each of you will need to drive a mile south, then a mile east, and then a mile north to get the gyroscopes into temporal alignment.
That would be easy, except each calibration requires placing a portable time gate at your starting point, then driving into it at the precise end of your three mile drive without it moving. Not only that, but your two time gates must be placed at least 100 miles apart so their signals don’t interfere. In other words, you need to find two different locations where you can drive a mile south, a mile east, and then a mile north and end up exactly where you started.
The professors are about to show you where you can do this when they vanish, becoming victims of the collapsing timeline. It’s up to you now: where can you place the time gates? Pause here to figure it out yourself.
If the earth were flat, there would be no way to solve this riddle. What you need is some way to use the sphericalness of the planet, some of its notable features, and the fact that lines of latitude aren’t really lines— they’re circles. The equator is the biggest of these circles, but it doesn’t do much for you, and you can’t even go south from the South Pole. But let’s try the North Pole. When you go a mile south, you’ll be on a circle that runs east west. After a mile east, you’re still on that same circle, so the final mile north brings you back to your starting point. Perfect. That’s one gate down, but where should you place the second?
Well, the nice thing about circles is that if you travel on them far enough, you come back to where you started. If there were a circle that had a circumference of one mile, that would work marvelously. You could drive south a mile to reach it, make the one mile rotation east, then go north a mile to return to where you started. Such a circle does exist just north of the South Pole. To find your starting point, you can use the standard formula for the circumference of a circle. If you wanted to be absolutely precise, you could use an equation that takes into account the roughly spherical shape of the Earth. But an area this small is so close to being flat that the standard formula gives a solution within a fraction of an inch of the actual distance. A circle with a one mile circumference has a radius of just under 0.16 miles, so any point on the circle one mile north of that will be suitable for your time gate.
In fact, there are other answers too: start a little further south, and you could travel east around the Earth twice, three times, or more! In theory, there are infinite possible starting points, but the circles get so tight that they aren’t actually practical to drive. Everything’s getting a little weird by the time you reach your starting points. You drop the time gates, sync up with your doppelgänger, and slam down the pedal. You both reach 88mph just as you complete your three mile circuits, merge the timelines, and save the universe.