Suppose I row out into a circular pond and suddenly fog falls. All sense of direction is lost (and I left my phone at home, so no help there).
What is my best strategy, in terms of (a) is sure to get me to shore and (b) get me there in the shortest distance? I don’t think this is obvious at first.
But now suppose instead I row out to sea from a straight coastline and the same thing happens; suddenly fog falls. All sense of direction is lost (and I left my phone at home, so no help there).
What is now my best strategy, in terms of (a) is sure to get me to shore and (b) get me there in the shortest distance?
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It's not even obvious at second - to me, at any rate. Help?
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For the circular pond we can find a strategy that will always succeed and cannot be improved on.
Suppose the radius of the pond is 1 km. Then rowing in a straight line must bring us to the edge of the pond in at most 2 km, because any line of length 2 km, which is the diameer of the circle, must intersect the shoreline.
This is the best possible. Suppose we chose some other path of length L < 2 km. Then whatever the shape of the path it could alway happen that the circle was centered at the mid-point of our path. Because no point on our path can be more than L/2 < 1 km from the midpoint and all the points on the circle are 1 km from the midpoint our path cannot intersect the shoreline.
The second problem is a lot harder. There is a solution which is thought to be the best but a totally convincing argument that it really is optimal seems elusive.
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h, OK. I was lost on a different path, trying to work out how you could know that you were travelling in a straight line in foggy conditions.
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The lost at sea version is discussed here (you'll have to live with some ads)
The problem has been around for a long time. The various candidates the video goes through are really interesting to follow but at the end of it all we still have no proof that we have found the best possible.
In fact there might not be a best possible. How so? Well there might be a number d that no solution can do better than, but no actual solution that attains this minimum. So there could be an endless series of minor improvements, each closer and closer to d but never reaching it.