# The lost drone at the Great Wall of China

A drone is stationary at a spatial point about 1 m from the Great Wall, which is a vertical plane rectangle with height 5 m and infinite length. The altitude of the drone is a random number between 0 m and 5 m. The drone may be pointing in any direction (which may even be tilted upwards or downwards), however it does not know which. It can travel in any direction, and switch directions accurately. It gets to know when it touches the ground plane, and can then determine its own tilt upwards/downwards by travelling a small distance on the ground (about zero).

Q 1: What is the minimum distance that must be travelled to ensure it hits the Great Wall?

Q 2: If the objective is to minimise displacement between initial and final positions, what upper value can it be guaranteed to outperform?

• A few points for clarification: (a) “about 1 m from the Great Wall”, and “a small distance on the ground (about 0)” — I guess we are to understand both these abouts to mean “to within some accuracy much smaller than all the other numbers involved”? (b) “a random integer between 0m and 5m” — by integer do you mean an integer number of metres, and (c) by random, do you mean “uniformly distributed random”? Commented Jan 21, 2016 at 16:11
• What is the maximum angle at which this drone can be tilted? Could "forward" ever be straight up? Commented Jan 21, 2016 at 16:58
• @Peter (a) Yes, that is right. (b) Thanks for pointing out; I've edited the q. (c) Solution should work for every scenario and not based on probability Commented Jan 21, 2016 at 17:28
• I wish this problem did not have the height restriction, and instead had a drone searching for an infinite plane. That problem (which I've thought about for years but haven't made good progress on), has the allure of a potentially elegant solution. But your height restriction probably precludes an elegant solution, which makes it a less attractive puzzle in my opinion. Commented Jan 21, 2016 at 18:05
• Making it an infinite wall also makes it fit into the generalization: How do you escape a $d$-dimensional forest if you know its size/shape and your distance from its boundary? Commented Jan 21, 2016 at 18:29