NavList:

Fortunately, the farmer has studied the origins of quantum mechanics and while reading about path integrals, he learned about the classical principle of least action and the optical equivalent, Fermat's Principle of Least Time (read about it in the Feynman Lectures). So turn the river into a mirror and the farmer into a light ray, and the problem is solved. If that's too hard to imagine, turn the river into a pool table bumper while the farmer becomes the cue ball at his house and the barn becomes the eight ball. Now hit the eight ball, rail first with no unusual spin on the cue ball. That's the shortest path.

The problem could be extended and the optical analogy continued by adding a marsh to the problem. Suppose there is marshy ground on either side of the river. On normal ground the farmer can walk at 4mph. On marshy ground the farmer can walk at 2mph. The change in speed is equivalent to refraction and shifts the fastest path towards the perpendicular to the river bank on the portions in the marsh.

Frank Reed