The recent expedition to the lost city of Atlantis discovered scrolls attributted to the great poet, scholar, philosopher Josephine.
logic/josephine
Problem
The recent expedition to the lost city of Atlantis discovered scrolls attributted to the great poet, scholar, philosopher Josephine. They number eight in all, and here is the first. THE KINGDOM OF MAMAJORCA, WAS RULED BY QUEEN HENRIETTA I. IN MAMAJORCA WOMEN HAVE TO PASS AN EXTENSIVE LOGIC EXAM BEFORE THEY ARE ALLOWED TO GET MARRIED. QUEENS DO NOT HAVE TO TAKE THIS EXAM. ALL THE WOMEN IN MAMAJORCA ARE LOYAL TO THEIR QUEEN AND DO WHATEVER SHE TELLS THEM TO. THE QUEENS OF MAMAJORCA ARE TRUTHFUL. ALL SHOTS FIRED IN MAMAJORCA CAN BE HEARD IN EVERY HOUSE. ALL ABOVE FACTS ARE KNOWN TO BE COMMON KNOWLEDGE. HENRIETTA WAS WORRIED ABOUT THE INFIDELITY OF THE MARRIED MEN IN MAMAJORCA. SHE SUMMONED ALL THE WIVES TO THE TOWN SQUARE, AND MADE THE FOLLOWING ANNOUNCEMENT. "THERE IS AT LEAST ONE UNFAITHFUL HUSBAND IN MAMAJORCA. ALL WIVES KNOW WHICH HUSBANDS ARE UNFAITHFUL, BUT HAVE NO KNOWLEDGE ABOUT THE FIDELITY OF THEIR OWN HUSBAND. YOU ARE FORBIDDEN TO DISCUSS YOUR HUSBAND'S FAITHFULNESS WITH ANY OTHER WOMAN. IF YOU DISCOVER THAT YOUR HUSBAND IS UNFAITHFUL, YOU MUST SHOOT HIM AT PRECISELY MIDNIGHT OF THE DAY YOU FIND THAT OUT." THIRTY-NINE SILENT NIGHTS FOLLOWED THE QUEEN'S ANNOUNCEMENT. ON THE FORTIETH NIGHT, SHOTS WERE HEARD. QUEEN HENRIETTA I IS REVERED IN MAMAJORCAN HISTORY. As with all philosophers Josephine doesn't provide the question, but leaves it implicit in his document. So figure out the questions - there are two - and answer them. Here is Josephine's second scroll. QUEEN HENRIETTA I WAS SUCCEEDED BY DAUGHTER QUEEN HENRIETTA II. AFTER A WHILE HENRIETTA LIKE HER FAMOUS MOTHER BECAME WORRIED ABOUT THE INFIDELITY PROBLEM. SHE DECIDED TO ACT, AND SENT A LETTER TO HER SUBJECTS (WIVES) THAT CONTAINED THE EXACT WORDS OF HENRIETTA I'S FAMOUS SPEECH. SHE ADDED THAT THE LETTERS WERE GUARENTEED TO REACH ALL WIVES EVENTUALLY. QUEEN HENRIETTA II IS REMEMBERED AS A FOOLISH AND UNJUST QUEEN. What is the question and answer implied by this scroll?
Show solution
The two questions for scroll #1 were: 1. How many husbands were shot on that fateful night? 2. Why is Queen Henrietta I revered in Mamajorca? The answers are: If there are n unfaithful husbands (UHs), every wife of an UH knows of n-1 UH's while every wife of a faithful husband knows of n UHs. [this because everyone has perfect information about everything except the fidelity of their own husband]. Now we do a simple induction: Assume that there is only one UH. Then all the wives but one know that there is just one UH, but the wife of the UH thinks that everyone is faithful. Upon hearing that "there is at least one UH", the wife realizes that the only husband it can be is her own, and so shoots him. Now, imagine that there are just two UH's. Each wife of an UH assumes that the situation is "only one UH in town" and so waits to hear the other wife (she knows who it is, of course) shoot her husband on the first night. When no one is shot, that can only be because her OWN husband was a second UH. The wife of the second UH makes the same deduction when no shot is fired the first night (she was waiting, and expecting the other to shoot, too). So they both figure it out after the first night, and shoot their husbands the second night. It is easy to tidy up the induction to show that the n UHs will all be shot just on the n'th midnight. The question for scroll #2 is: 3. Why is Queen Henrietta II not? The answer is: The problem now is that QHII didn't realize that it is *critical* that all of the wives, of faithful and UH's alike, to *BEGIN*AT*THE*SAME*MOMENT*. The uncertainty of having a particular wife's notice come a day or two late makes the whole logic path fall apart. That's why she's foolish. She is unjust, because some wives, honed and crack logicians all, remember, will *incorrectly* shoot faithful husbands. Let us imagine the situation with just a SINGLE UH in the whole country. And, wouldn't you know it, the notice to the wife of the UH just happens to be held up a day, whereas everyone else's arrived the first day. Now, all of the wives that got the notice the first day know that there is just one UH in the country. And they know that the wife of that UH will think that everyone is faithful, and so they'll expect her to figure it out and shoot her husband the first night. BUT SHE DIDN"T GET THE NOTICE THE FIRST NIGHT.... BUT THE OTHER WIVES HAVE NO WAY OF KNOWING THAT. So, the wife of the UH doesn't know that anything is going on and so (of course) doesn't do anything the first night. The next day she gets the notice, figures it all out, and her husband will be history come that midnight. BUT... *every* other wife thought that there should have been a shooting the first night, and since there wasn't there must have been an additional UH, and it can only have been _her_ husband. So on the second night **ALL** of the husbands are shot. Things are much more complicated if the mix of who gets the notice when is less simple than the one I mentioned above, but it is always wrong and/or tragic. NOTE: if the wives *know* that the country courier service (or however these things get delivered) is flaky, then they can avoid the massacre, but unless the wives exchange notes no one will ever be shot (since there is always a chance that rather than _your_ husband being an UH, you could reason that it might be that the wife of one of the UH's that you know about just hasn't gotten her copy of the scroll yet). I guess you could call this case "unjust", too, since the UH's evade punishment, despite the perfect logic of the wives.