An Asteroid Walk: The Answer

In 2016, I posted a cute puzzle, An Asteroid Walk, on my blog. Here it is. Puzzle. Seven astronauts landed on a small spherical asteroid. They wanted to explore it and walked in different directions starting from the same location. All used the same walking algorithm: walk x kilometers forward, turn 90 degrees left and walk another x kilometers, turn 90 degrees left again and walk the last x kilometers. The value of x was different for different astronauts and was one of 30, 40, 50, 60, 70, 80, and 90. All but one astronaut finished in the same location. What was the value of x for the astronaut who finished alone? What is the size of the asteroid? To be precise, by size, I mean the great-circle circumference. I never posted the answer. This post had sat on my to-do list for more than ten years, and I finally got around to it. The answer is surprising: the value used by the lonely astronaut is uniquely determined, but the size of the asteroid is not. Let c be the circumference of the…