You are driving along a perfectly straight road through the woods and decide the trees look like an inviting hike. After all, you have a GPS that could easily get you back to the road. So you get out and head off in a straight line perpendicular to the road, not paying any attention to your direction because, hey, GPS.

Unfortunately after traveling 1km your GPS crashes. It loses all of its map data as well as any previous journeys. In you panic, you even forgot which direction you were walking.

So here you are: 1 km away from the long straight road (the only one around for many km) in an unknown direction. You have a GPS that can still give your accurate position and path relative to your start.

What is the length of the shortest path (measured from here) that guarantees you will find the road?

Note: the trees are dense enough that you could be very close to the road and not see it.

I've tried this. Inscribe the circle on a regular polygone, and consider that one of the sides is the road. If the road do not match one of the sides you would anyway find the road before completing the path, that I'm going to describe. You walk from the center of the circle to a vertex of that side of the polygon chosen as road. From there to the circle following the side of the vertex which is not the road. Once arrived to the circle (it happens in the middle of the side) follow the circle since the middle of the side whose last vertex is the other vertex of the road. From there leave the circle and follow the side till you find the vertex and also the road. That would be the minimum path for that polygon.
It's not difficult to build a general formula for the distance d for a polygon with n sides [d=d(n)].
Using it this are my results.
N=3 d=7.55
N=4 d=6.55
N=5 d=6.44
N=6 d=6.49
For higher values d increases. So if I made no mistakes it is posible to walk just 6.44 km, using a pentagon as model and be sure you find the road
But perhaps working other methods it's posible to refine 20 or 30 metres, I don't know.

















Edited on January 9, 2016, 5:35 pm

Posted by armando on 2016-01-09 11:01:26