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A dark circular forest has diameter 10 miles. I drop you off somewhere blindfolded, then you can see again. You can make a plan how to walk on a sheet of paper (it does not have to be straight lines), which plan you can exactly accomplish with a GPS. The GPS cannot tell you how to go out from the woods though.

a) What is the smallest positive real number $k$ with the property that you can surely exit the forest by following the path of your plan whose distance is $k$? And what is that path? Is it a weird curly curve/spiral or just some straight line segments? (Note that $k\le 10$ is obvious. Can you do better than 10?)

b) If you are dropped inside the circle with uniform distribution, what is the best plan to minimize the expected value of the distance of your walk?

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closed as off-topic by JonMark Perry, Claude Leibovici, Did, Brian Borchers, Narasimham Jan 20 '18 at 18:45

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    $\begingroup$ Pretty sure the straight line of length equal to the diameter is minimal (If you had a path of length less than the diameter, place it so it's midpoint is at the center...pretty clear it can not meet the circumference). In any case, here is a good summary of some of what is known. $\endgroup$ – lulu Jan 20 '18 at 0:41
  • $\begingroup$ I feel like some sort of spiral pattern that lets you make a full circle in less than 10 miles of walking might do it; that way, if you're on the edge of the circle and start in the opposite direction, you spiral back after a maximum of 5 miles, and if you are in the middle, you still get out in 10 miles, but I don't know what angle that would be if possible. $\endgroup$ – Davy M Jan 20 '18 at 1:47
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    $\begingroup$ My intuition is also telling me you won't beat a straight line, with direction chosen randomly. Since the circle is symmetric, any bias will be better and worse in equal proportions. $\endgroup$ – dbx Jan 20 '18 at 17:34