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?