# The river shore problem

I have a one strange question. Hope that it'll be fine here. Imagine we have a moving point in an infinite length and 2 meters width corridor. This point needs to get to one of the borders of a corridor and reach it. At the first moment of time, our dot is located somwhere in that corridor, it's not stated, where exactly. The question - Is there a path which certainly leads to the wall, for which the maximum distance between the first location of a point and the border will be not more than

a)$483$ sm

b)$462$ sm

c)$458$ sm

By the way, we consider our corridor is a one dark place, so our point can not see where to go.

I came up with an idea that our point's trajectory will be circular. That's guaranteed that a point will reach the border no matter where it starts. But I can't figure out how to use this restrictions to solve the problem. Help me please.

Of cource the question is not just to tell whether there is such path, but also try to figure out a trajectory equation.

Sorry for tags, don't know what to put here.

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The problem is solved by Ani Adhikari and Jim Pitman, The shortest planar arc of width 1, American Mathematical Monthly 96 (1989) 309-327. If you can't access that paper, another proof of the Adhikari-Pitman result is given here.

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Thanks. I'll read about it. –  user1131662 Jan 22 '12 at 2:11
Well, the width is 1, hope I can work out the solution based on this document for 2 m. width. –  user1131662 Jan 22 '12 at 2:31