Case 1: When there is only one UH in the kingdom his wife doesn't know of any cheaters, so the queen's announcement
is a revelation to her. She kills her husband the next midnight. All good!
Case 2: When there are only two UHs in the kingdom their respective wives know of only one cheater and incorrectly
think that they are in the case 1 and the queen's announcement
will be a revelation to the other cheater's wife. When nothing happens on the first night they now know that their husband is a cheater too and kill their husbands the next midnight. All good!
Case 3: When there are three or more UHs in the kingdom there is
nobody who thinks they are in the case 1. In other words, no wife thinks that there is one UH. No wife will expect anything to happen on the first night or any other night because the queen's announcement is
not news to anybody and they
all know that it's not news to anybody.
Here are the contradictions in the explanations given in this thread:
yes - I repeated the solution, since I assumed that it is clear ... I will try to take it a step further ...
so let's assume that I am a wife and I know that there are 3 UH's
1st night
Let's see if all 3 UH's will be shot ... but nothing happens ... I could have known that ... I guess that each of the 3 wives expected 2 shots ... hmmm, maybe the next night
2nd night
let's wait what happens ... I guess that each of the 3 wives is anxiously awaiting the 2 shots now ... I imagine that one of them might think as follows: "I know there are 2 UH's. They were not shot the first night, since both of the 2 wives thought that there is just 1 UH. So this 2nd night both of them must kill their UH's. What the ...? I hear no shots. And now I know why - not only those 2, but also I do have an UH. We'll shoot them the 3rd night" ... I think that is the way one of the 3 wives might have thought
3rd night
I'll open the window ... time is ticking away and then I hear something ... nope, was just some cat outside ... what is going on here? no shots? it is morning already ... the wives must have thought the way I imagined it ... unless ... wait a minute ... each of them does not know 2 UH's but 3 of them ... so there are 4 UH's? ... but who is the 4th one???
oh noooooo ... where is my gun ...
The highlighted part of the logic is faulty because everybody knows at least 2 UHs and everybody knows that everybody knows that there are at least 2 UHs.
I guess it's hard to grasp this answer unless you understand the simple fact that each wife knows every other husband's status but their own. Or, phrased another way. If I am a wife of Mamajorca, every other woman in the town knows whether my husband is unfaithful or faithful. So, from my perspective, if I know that 20 husbands are unfaithful SOBs, then I will anxiously await the 20th night to hear the gunshots.
If my husband is faithful, then there really are only 20 unfaithful men in the town. This means that 20 wives in the town will be believing that there are only 19 unfaithful husbands and would be waiting for the 19th night instead. When the 19th night has come and gone with no gunshots fired, then they will know that their husbands are unfaithful and will wait for midnight on the 20th night to shoot them. I will know that my husband is indeed faithful and probably breath a sigh of relief.
If my husband is unfaithful, then there are really 21!!! unfaithful husbands, the 20 that I know of plus my own. So, again, I still wait for the 20th night, hoping that gunshots will be heard (from the women who would have believed 19 unfaithful husbands exist), but none would come, because instead there are 21 women waiting for the 20th night (believeing there were 20 unfaithful husbands) and did not shoot their husbands. All of those 21 women will now know their husband is the unfaithful one, because they know all the other women in town are thinking of 21 unfaithful husbands (the 20 that I know about plus my own husband). So, on midnight of the 21st night, those 21 women (including me) will shoot their husbands, knowing, full out, that those husbands are unfaithful.
Long winded! Wow.
This logic recursively
assumes that the other women will be waiting and counting the nights. Why would they do that? They haven't learned anything new from the queen's announcement. This logic relies on the assumption that this logic is correct to prove that it's correct.
I have been struggling with the solution for a while and I was about to post my opinion on the fact that critical information was missing, but then I re-read and realised...
Take an Arbitrary number of couples (singles are irrelevant). Let's choose 250 Married Couples.
We know there are 50 UH (Unfaitful Husbands).
Each W (wife) only knows of the total number of OTHER UH's, not the ACTUAL number of UH
So...
200 W believe there are 50 UH (but they don't know if there are 51)
50 W believe there are 49 UH (but they don't know if there are 50)
0 W believe there are less than 49H
The reason for the (nested) inductive logic is as follows...
The 200 W believe there are 50 UH and believe that each of those 50 W believe that...
There are 49 UH and will shoot their husbands on night 50 and that those 49 W believe that...
There are 48 UH and will shoot their husbands on night 49 and that those 48 W believe that...
There are 47 UH and will shoot their husbands on night 48 and that those 47 W believe that...
There are 46 UH and will shoot their husbands on night 47 and that those 46 W believe that...
....
There are 2 UH and will shoot their husbands on night 3 and that those 2 W believe that...
There are 1 UH and will shoot their husbands on night 2 and that those 1 W believe that...
THERE MUST BE AT LEAST 1 UH
So as the nights increase the woman see that logic unravel.
When it does reach night 50 and there is no shots, ALL 50 W realise that there was no-one out there that believed there were 48 UH.
If someone believed there were 48 UH they would have shot their husband on night 49 because the logic dictates that if my husband is Faithful, then 49 W must have believed there were 48 UH, and if no-one believed there were 48 UH, then all 50 W realise their Husbands are cheaters and shot them the next night, night 51.
It works because all woman are perfectly equal in their logic. Although at night 50, the 200 W might have started sweating and been extremely overjoyed by the mass murder on night 51. 
The highlighted statement is true, but forgotten in the explanation that follows. If all know that no wife believes that there are less than 49 UH then the logical chain breaks.
This topic is locked
