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334 logic/josephine.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

334 logic/josephine.p


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?

logic/josephine.s

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.

 

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