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 degree 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?
(Konstantin Knop)
Walking on the surface of an asteroid implies the existence of a local g force, pointing to the center of the sphere, and the walk must be therefore perpendicular to the local surface of the sphere, which means moving along a great circle.
"forward" means remaining in the same plane of the great circle. Any other movement, for example along a "latitude" would necessitate a constant turning so as to get out of the great circle.
Under the above understanding- if the circumference of the sphere would comprise a common product of the different values of x, then each of the astronauts would have completed a full number of walks around the asteroid and returned to the original location. This would occur if the circumference would have been 1,2,5, or 10 Km.
Another possibility to reach the same end point would have been if all the astronauts would walk the
same distance
x = 0.25*circumference, even if they started out in different directions.
None of the above, answers the conditions stated in the problem, and I wonder if a solution exists which adheres to the above understandings.
Meaning of "walking forward upon the surface of a sphere
