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)
There is some ambiguity in the formulation of this puzzle. Charlie has hit it in his post.
The question is: the astronauts begin to walk forward (I assume that they follow different great circles each of them), then turn 90 degrees left an in this point they can do two different things: they can follow a parallel or they can follow a great circle. The problem only say "turn 90 degrees left and walk another x kilometers". Both possibilities are open with that formulation.
If they follow a parallel: all of them are going to come back to the starting point except if their walk is half the size (or a multiple like 3/2, 5/2, ... of the size) of the asteroid. It means that there are different solutions for the problem in this case (for ex: the size could be 100, 120, 140, 160, 180 km: all of them satisfy the problem).
If instead they follow a great circle after the first turn, they are not going to come back to the starting point nor to finish in the same point their path, except if on the second leg all of them moves along the same great circle or if they finish every leg (and so also the whole path) in the starting point.
This can occur if the size of the asteroid is 40 km, because in this case the astronauts walking 30, 50, 70 e 90 km walk on the same great circle (like the ecuator if the starting point is like North Pole). At the end of their paths they will be on the starting point. The astronauts walking 40 and 80 km finish all the legs of their path (and so their whole path also) in the starting point.
The astronaut walking 60 km finishes instead in the opposite point of the sphere.
Edited on April 21, 2016, 4:54 pm

Posted by armando
on 20160421 13:56:04 