David Aberle
Other correct solutions were sent in by Ray Kremer, Andrew Hess. Further solutions were submitted by Ivan Lisac, Brian Laughlin, Philippe Fondanaiche, Burkart Venzke,
Refer to the diagram on the left. We start by
assuming, as we may, that the length of AB is 1. It's easy to
see that the endpoints of each successive line segment (after the first)
bisect the two line segments they sit on; for example, the endpoints
D and E of second line segment bisect BC and
AC, respectively. Using this observation, and analyzing the
horizontal and vertical components separately, you can see that your
first move, from A to D, takes you 1/2 unit forward;
the second, from D to E, takes you 1/2 unit backward;
the third, from E to F, takes you 1/4 unit forward; the
fourth does not change your horizontal position; the fifth takes you
1/8 unit backward and so on. Thus your limiting horizontal
position is represented by the sum
This last sum is a geometric series with ratio -1/4. A similar analysis of the vertical component gives you that your limiting vertical position is 2/5.
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