I was a little surprised at the length of the comment thread in the post about the logic puzzle involving the monk Gaito going up and down a hill. On the one hand, I thought that there were some excellent explanations of why there had to be at least one instant where the monk was at the same location at the same time. These involved visualizing the situation in slightly different ways, such as instead of having one monk go up and down on two different days, having two monks going up and down on the same day or using graphs or films and so on.
But clearly these arguments were not persuasive enough for some and I have been trying to think why this might be so. In my teaching experience, it is often the case that what seems obvious to you as a teacher is by no means so to the student. It is no use repeating the same explanation more slowly or (worse) more loudly or (much worse) exasperatedly. There is clearly some opposing argument that the student finds persuasive that makes them reject your argument and yet they may not be able to identify and articulate what it is. Instead they feel that there must be some flaw in your reasoning that they cannot put their finger on. It is more fruitful as a teacher to try and figure out what their argument might be, rather than reiterating your own.
Comment #56 by larpar is the clue that there is something like that going on here when they say “But something in the back of my head, something I’ve heard before, tells me there is something wrong with comparing simultaneous and consecutive events. Sorry I can’t be clearer.” Also the comment #68: “I have considered being wrong. I even admitted being wrong about one aspect of my reasoning, but I think I have resolved that error. Maybe not. I could still be wrong. One other reason that I’m not yet ready to admit defeat is that this is a “logic problem”. Usually in these types of questions there is some kind of twist that makes the obvious answer wrong. I’m holding out for the twist. It may never come. : )”
In such situations one has to try and identify and put into words their argument. In this case, I think it may be because at the back of their minds is an argument along the following lines: If you take any given point on the path, there is no guarantee that the monk will be at that particular point at the same time on the two journeys. This is easy to see since all we have to do is vary the speeds of the monk during the two journeys so that that point is reached at different times. In other words, the probability that the two journeys will coincide at any given point at the same time is of measure zero. This can be repeated for every point on the path, because we can always postulate walking speed scenarios where the two journeys will reach that point at different times. It is a short step to go from there to think that there is no point at which this is guaranteed to happen. This argument is a powerful one that has significant plausibility. Before I present the subtle flaw with it, I would suggest that readers think about how they might counter it.
The flaw is that there is a difference between picking individual points in advance and arguing that the simultaneity of time and place cannot happen at any of them, to arguing that it cannot happen at any unspecified point. As an analogy, I am planning to leave my home sometime later today. If I specify an exact time, the probability of my leaving exactly at that time is zero, irrespective of what time I pick. But the probability that I will leave at some time is one. i.e., it is guaranteed. (Let’s leave out the possibility that something might cause me to change my plans about going out.)
This is a subtlety that arises with continuous distributions, like the monk’s position along the path or the time at which I leave. If we are dealing with discrete distributions, then the situation is different and their argument holds. For example, if instead of a path, the monk had to navigate 1000 steps while going up and down the the hill. Then there is no guarantee that he will be on the same step at the same time on the two journeys, because the time at which they cross could occur when they are each moving from one step to the next. i.e., the two paths could cross when the monk is in the process of going from one step to the next one up on the upward journey while taking a step down on the downward journey, so that they are not on the same step at the same time even though they are crossing. (This is the point that Deepak Shetty seems to be hinting at in comment #27.)
This long response is only partly about the monk puzzle. It is also about (a) the difference in probability estimates between continuous and discrete distributions of outcomes and (b) how important it is to not dismiss those who cannot seem to see what seems obvious to us, because understanding the reasons for their skepticism can reveal important insights about subtleties.