My take:
a) If you interpret it to mean that our hiker will arrive at the same rock at the same time on both the trip going up and the trip going down - then no, there is no way to prove that he would hit that same spot at the same time on two consecutive days. (I.e. he will not necessarily stop and have lunch at 12PM at the same rock on both days.)
b) But if you looked at his progress the previous day superimposed over his current progress (they are heading in opposite directions), then the hiker and his previous day's "shadow" would occupy the exact same space on the trail at some time during the trip down. No one knows what point on the path that will be and no one knows what time that will be, but we do know that it will happen. I think that is all Kotysdad was trying to show with this problem - feel free to correct me if I am wrong, I won't be offended.
