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Asteroid walk (Posted on 2016-04-19) Difficulty: 3 of 5

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)

See The Solution Submitted by broll    
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an possible solution | Comment 3 of 8 |
I hope this is not totally wrong. 
Think about this scenary: 

  • The path of each astronaut begin along different great circles from the starting point (I mean lines in planes containing the centre of the sphere).
  • The great circles of the sphere are 120 km long. 

Let see what happens with each astronaut:

  • The astronaut walking 30 km (A) walk a quarter of a great circle, turns left to do another perpendicular quarter and then turns again left for other 30 km. She is moving along a spherical triangle whose sides are the interseccion of the sphere with three orthogonal planes (as like delimiting axes x, y, z) centered in the center of the sphere. 
  • Same thing for the astronaut walking 90 km (G). If he starts walking on the opposite direction of A (and three times faster) they will meet at every turn.

  • The astronaut walking 40 km (B) begin her walk along another great circle. She walks 10 km more than a quarter of great circle. and then turn left along the parallel line delimited by an orthogonal plane (as if she changes from a meridian to a parallel on the Earth). She walks 40 km along the parallel and then turn left to come back to the starting point. 
  • Same thing for the astronaut walking 80 km (F). If he begins walking in the opposite direction of B they could meet two times before their meeting at the final point. 
  • Same thing for astronauts walking 50 km (C) and 70 km (E).

  • But if the astronaut walking 60 km (D) begin walking along a great circle, she will be in the opposite point of the sphere after the first walking, then she turns left and will be again at the starting point, and then left again to finish at the opposite point of the sphere. 

Edited on April 20, 2016, 9:23 am
  Posted by armando on 2016-04-20 09:06:01

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