NavList:

Brad, you wrote:
"Requirements: two people, one string, no cup."

Ahem. A thousand yards of string??! I agree it will work in principle, but it's not practical. A scale model is the way to go, and it performs the optimization automatically. And I want to emphasize that the string in the scale model that I described earlier, if it's free to slide easily along the dowel, will find the optimal solution with excellent accuracy, just as a string pulled across a globe of the Earth, if it is free to slide, will find the proper great circle path over the Earth.

There's a variant on this puzzle that has a nice "image" solution and cannot be easily modeled by taut strings. Suppose the farmer walks twice as slowly or four times more slowly when burdened with water. That's a reasonable scenario. So as before, we look at the image barn "reflected" on the other side of the river (the river assumed to be nothing more than a line). On the house side of the river, the farmer walks at 4mph. On the opposite side, he walks at 2mph or 1mph carrying water. And this gives us a simple refraction problem with the index of refraction equal to 1 on the house side and 2 or 4 on the image side. The angles of incidence, a, and refraction, b, are then given by Snell's Law: sin(a)/sin(b)=2 (or 4). Back on the non-image side of the river, this implies that the farmer would walk somewhat further north to the riverbank before turning and being slowed down by carrying the water to the barn. And it's all perfectly calculable.

Frank Reed