You are hiking in a half-planar woods, exactly 1 mile from the edge, when you suddenly trip and lose your sense of direction.

geometry/hike Added material

Problem

You are hiking in a half-planar woods, exactly 1 mile from the edge,
when you suddenly trip and lose your sense of direction.  What's the
shortest path that's guaranteed to take you out of the woods?  Assume
that you can navigate perfectly relative to your current location and
(unknown) heading.

Show solution

Go 2/sqrt(3) away from the starting point, turn 120 degrees and head
1/sqrt(3) along a tangent to the unit circle, then traverse an arc of
length 7*pi/6 along this circle, then head off on a tangent 1 mile.

This gives a minimum of sqrt(3) + 7*pi/6 + 1 = 6.397...

It remains to prove this is the optimal answer.

Additional material

[ Editor: On the proof of optimality: Lawrence Flynn writes that the
proof of the minimality is presented in David Karr's response to
Flynn's rec.puzzles posting "Flying saucer spy plane revisited" (a
thread of two posts).  Flynn adds that he can also provide geometric or
calculus proofs of the minimality ("granted one must add that the
shortest route from the center to a circle is a radius").  This is a
paraphrase of Flynn's 1998 mail to Chris Cole about "finding a cliff";
the proof itself is not in the collected material. ]

-- Lawrence Flynn <leflynn@my-dejanews.com>