Math puzzle: The homesick rover

Far away, on a large, rocky exoplanet, a rover from Earth has just arrived on a mission of exploration. But the poor thing doesn’t want to explore. The moment it begins moving, it grows homesick and wants nothing more than to return to the very spot where it began. Unfortunately, it has received very explicit instructions. The first day, it must travel one kilometer forward in a straight line, then turn 90 degrees. The second day, it must travel two kilometers forward, then turn 90 degrees. The third day, it must travel three kilometers forward, then turn 90 degrees. The fourth day, it must travel four kilometers forward, then turn 90 degrees. And so on, for the duration of an eight-day mission.

The rover wants to return to the site where it landed. But it can choose only which direction to turn, left or right, at the end of each day. How can the rover end the eight-day mission back at its landing site?

BONUS: A hundred rovers with the same set of instructions have been sent to other large, rocky planets, each with a different mission length, ranging from one day to 100 days.

How many of these rovers can end their missions back at their landing sites?

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