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?

There may be a better way then straight back and forth.
I just sketched it out, somebody smarter will have to do the math.
Start at the center of the square, mowe under a 45 degree angle (parallel to the diagonal).  Once you reached the side of the area, turn another 45 degrees.  Continue doing this (like a ball on a billiard table), but at the edges continue a bit outside of the square, such that after your next 45 degree turn, the following lane is just next to a lane you have already mowed.
This should be better then straight back and forth because you also need 90 degrees to mowe in the opposite direction, but you cover more ground by zig zagging then by going in a straight line.

Steve Herman: I liked the idea of going around the world and I hope you get paid by the hour.

Posted by Hugo on 2005-02-14 20:13:15