An area in the shape of a square 10 units on a side needs to be mowed. The mower, which only goes forward, has a mowing "footprint" that is a unit square, and turns about the center of its footprint.

An optimal mowing plan is sought. A mowing plan designates a starting position and from there gives a complete mowing path. An optimal mowing plan is one closest to a straight line in the sense that the sum of all the changes in the mower's angular direction is minimized, each such change taken as the minimum possible positive value. The mower is not impeded by the border of the square and can travel without difficulty outside it as well as inside it.

Is "spiral" better than "back and forth," and what about a "diagonal" plan?

(In reply to More efficient mowing by Hugo)

I don't think that the 45 degree scheme does it.  The diagonal is 14.14 units long, so I think that you will need to do 7 full 360 degree turns to get it all mowed.  The 45 degree spiral scheme mows a square that has a 14.14 unit side.

Hugo: Thanks for kind wishes.  I am a consultant, and I do get paid by the hour.  Plus, I've always liked to travel, so the around the world scheme appeals.  Now, if only Bill Gates has a 10 unit lawn that needs optimum mowing.  Another practical problem, though, is that the grass would probably be growing faster than I mowed it.

Posted by Steve Herman on 2005-02-15 15:02:29