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Thanks, Dave. I looked at his CV, and it's enough to know he's a physicist. That pretty well guarantees that this problem is based in optics, and more generally the foundations of quantum mechanics. It looks like a math puzzle, but it's built on the math at the heart of fundamental physics.

Here's another variant on all this. Suppose I'm the farmer's neighbor. I want to mess with him and arrange that all paths to the river end up taking the same amount of time. We'll leave the barn out it. All we're interested in now is sending the farmer to the river to get a bucket of water, but we're going to arrange the marshlands in such a way that no matter which path he takes to the river (within some range), it's always the same amount of time to get to the water (and back, but we can ignore that, and we'll also ignore intentional delays like walking around in circles). The riverbank is a straight N/S line 400 yards east from the house. A zone along the riverbank of variable width is marshy and slows the farmer's walking pace. Clearly if we go straight towards the river, perpendicular to the riverbank, that will give us a standard to compare against. At that spot on the riverbank, the marshy zone is 40 yards deep. But suppose the farmer notices that the marsh is somewhat thinner one either side of this path. It might be faster to head to the river at a slight angle... So he starts out towards the river the next day on a path at a small angle, let's say 2° off-axis, from the previous day. He walks in a straight line and sure enough the marsh isn't quite as deep, but lo and behold the additional path length has exactly cancelled out the gains from the narrower marsh. The next day he doubles the offset angle. The marsh is thinner still, but again the gain from less time in the marsh is offset by the increase in the distance before reaching the marsh. It cancels out. No matter what offset angle he tries, the paths take exactly the same amount of time (up to some limit of offset angle... beyond that, they're just longer)! Very frustrating. So finally our farmer gets in his hot air balloon (ok, he's a clever farmer) and looks down on the shape of the marsh zone along the riverbank. He sees that It is very nearly a segment of a circle extending out from the riverbank, and he realizes that all the paths he has tried will necessarily take the same amount of time to reach the water. Why is this? In optical terms, what have we made? What sort of optical element yields a bunch of paths with identical travel times from some fixed point? It's a mud and marsh version of a...??

Frank Reed