Yes. Let's start with the simple case that N = 1. The offended spouse reasons as follows: the priest knows there is at least one adulterer, but I don't know who this person is, and I would if it were anyone other than me, so it must be me. What happens if N = 2? On the first Sunday, the two offended spouses each calmly wait for the other to get up and condemn their spouses. When the other doesn't stand, they think: They do not think that they are a victim. But if they do not think they are victims, then they must think there are no adulterers, contrary to what the priest said. But everyone knows the priest speaks with the authority of God, so it is unthinkable that he is mistaken. The only remaining possibility is that they think there WAS another adulterer, and the only possibility is: MY SPOUSE! So, they know that they too must be victims. So on the next Sunday, they will get up. What if N = 3? On the first Sunday, each victim believes that the other two will not get up because they did not know about the other person (in other words, they believe that each of the two other victims thought there was only one adulterer). However, each victim reasons, the two will now realize that there must be two victims, for the reasons given under the N = 2 case above. So they will get up next Sunday. This excuse lasts until the next Sunday, when still no one gets up, and now each victim realizes that either the priest was mistaken (unthinkable!) or there are really three victims, and I am ONE! So, on the third Sunday, all three get up. This reasoning can be repeated inductively to show that no one will do anything (except use up N - 1 excuses as to why no one got up) until the Nth Sunday, when all N victims will arise in unison.

The induction can also be run "top down" on the priest's statement. What must everyone believe about what everyone else believes? N victims of adultery believe there are N - 1 victims, and that all of

these

N - 1 victims believe that there are N - 2 victims, and that all of

these

N - 2 victims believe that there are N - 3 victims, and that all of

these

...

2 victims believe that there is 1 victim, and that this

1 victim believes there are no victims.

Suppose the priest says, "There are N adulterers in this town." Then all the adulterers will know immediately that their spouses have been unfaithful, and will denounce them the next Sunday. Now, suppose the priest says, "There are at least N - 1 adulterers in this town." On the first Sunday, the offended spouses all wait for each other to stand up. When none do, they all know that they have all been horribly mistaken, and they stand up on the following Sunday. But what if the priest says, "There are at least N - 2 adulterers in this town." On the first Sunday, the victims reason, those N - 1 victims are going to be surprised when no one stands up, and they'll stand up next Sunday. On the second Sunday, the victims reason, wait a minute, since they didn't stand up, I must be one of the deluded victims. And everyone stands up on the third Sunday. This resoning applies inductively, adding one Sunday at each step, until the priest's original statement is reached, which takes N Sundays for everyone to unravel.

By the way, the rest of the town, which thinks there are N adulterers, is about to conclude that their perfectly innocent spouses have been unfaithful too. This includes the adulterous spouses, who are about to conclude that the door swings both ways. So the priest is playing a dangerous game. A movie plot in there somewhere?

This problem has been traced back to Puzzle-Math, George Gamow and Marvin Stern, Viking Press, New York, 1958, p. 20, attributed to Dr. Victor Ambarzuminian.

This problem is an analogue of the "dirty children" problem discussed in the seminal paper on common knowledge by Halpern and Moses (JACM 1990).

If the information of each victim is less than perfect, the problem is related to the "frame" problem of AI, cf. J. M. McCarthy? & P. J. Hayes, "Some philosophical problems from the standpoint of artificial intelligence" (1968) in Readings in Artificial Intelligence (pp. 431-450), Tioga Publishing Co., Palo Alto, CA (1981).

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