Puzzle 61 in R. Smullyan's _What is the Name of This Book?_ is about Tweedledum and Tweedledee.

[ Editor: posed by Col. G. L. Sicherman <gls@hrcms.ATT.COM>. ]

If it is not Sunday then the answer can only be given by the owner:
If it is one of his lying days he will lie and claim he is telling
the truth - and if it is one of his truthtelling days he will state
truly that he is telling the truth. The non-owner on the other hand,
would on a lying day tell that the owner lies (when he is actually
telling the truth) and on a truth-telling day tell that the owner
lies (which he actually does).

However on Sundays both will tell the truth and could have given
Alice's answer.

Humpty Dumpty is assuming that there is an equal (1/7) chance for
each day and an equal (1/2) chance to meet either T-dee or T-dum,
thus a 1/14 chance to meet the non-owner on a Sunday - hence 13/14
chance to meet the owner.

As anyone will know who has eaten pancakes with a gameshow host who
has two children out of which one is a girl, the puzzlitically correct
answer is 7/8 .
Under the assumptions we have 14 equal probability outcomes, out of which
we can discard 6 as not relevant. Of the remaining 8, in only one will
Alice have met the non-owner [Editor: "the owner" in the original].

-- Stein Kulseth <stein.kulseth@nta.no>

Before Alice hears the statement, there are 14 equally likely
possibilities:
   - it can be one of the seven days of the week (Alice doesn't know
     which)                                                 7
   - the brother can be the owner of the rattle or not    x 2

The speaker's statement eliminates six of the possibilities (the
non-owner could not make the statement on Monday through Saturday).

That leaves eight possibilities: the speaker is the owner (on any of the
seven days) or the speaker is the non-owner (on a Sunday).  So the
chance the speaker was the owner is 7/8.

Smullyan's error was to assume that the conversation was equally likely
to have happened on any of the seven days.  But in fact the statement
provides information about the day of the week as well; it was twice
as likely to have happened on a Sunday as on any of the other days,
because on Sunday either brother could have made the statement.

Interesting.  I certainly didn't catch that when I read the book.

-- Pete Kimball <U31850@uicvm.uic.edu>

In article <94031.034333U31850@uicvm.uic.edu>, Pete Kimball wrote:
>That leaves eight possibilities: the speaker is the owner (on any of the
>seven days) or the speaker is the non-owner (on a Sunday).  So the
>chance the speaker was the owner is 7/8.

Hmmm... This seems to suffer from the same sort of ambiguity as the two
daughters. We don't really know the likelihood of his answer given the
various possibilities, so it is difficult to apply Bayes rule. In
particular, one wonders what the answer would have been had she come
across the wrong brother on a non-Sunday. Perhaps "The non-owner of
this rattle is telling the truth today" is the most natural response.
But then either brother could make either statement on Sunday.

With this interpretation, a possibility is described by these parameters:

is owner?
preferred statement on Sunday
day of week alice asks question.

Each of these has a priori equally likely values. So there are 28
possibilities. His answer eliminates 6x2+2=14 possibilities. In 13
of the remaining possibilities Alice is speaking to the owner. 13/14.

-- David Wayne Ring <dwr2560@tamuts.tamu.edu>

I agree with the general point, which is:  We know nothing about the
actual probabilities of the statement being made, conditional on
day of the week and whether the speaker is the owner.  All that the
statement does is to EXCLUDE the possibility that it is the non-owner
speaking on a weekday.  That is, the probabilities of the
statement being made are:

              S     M     T     W    R     F     S
Owner         p11   p12   p13   p14  p15   p16   p17
Non-owner     p21   0     0     0    0     0     0

And we really don't know what the non-zero p's really are.  I concede
that the assumption which I, and Smullyan, and Humpty Dumpty apparently
made was that they are equal, or even 1 -  as if Alice supplied the
statement and asked "True or False?" (on the assumption that
Tweedledum and Tweedledee always answer such questions).
There is no solid ground for this premise.

Now, DWR supplies an alternative solution where p11 = p21 = .50
and the other p values are = 1.  He has the right to do so, but of course
this is on even shakier ground than the original.  In reality you
can come up with any set of probabilities you like based on imagined
propensities of Dum and Dee:  Dum has a 50% chance of napping if you
meet him on a Tuesday, non-owners of rattles never give informative
answers on a Friday, the owner of the rattle always goes home and plays
with it on a Sunday and isn't around to answer your questions.  DWR's
possible world, in which, on Sunday, both the owner and the non-owner
are as likely to make the "Ring statement" as the "Smullyan statement"
is one element of an infinite set.  Another element is the one in
which p11 = p12 =  .. p17 (that is, the owner makes the Smullyan
statement every day, for the sake of consistency) and p21 = 0
(the non-owner makes the Ring statement every day, for the sake
of consistency) in which case the speaker is the owner for sure.

However, since we really don't know which set of p's is right, or how
they might vary, isn't it best to treat them all equally?  In that case,
7/8 is still the best answer.

-- Pete Kimball <U31850@uicvm.uic.edu>

[ Editor: further material on this puzzle, edited separately from other collected postings (batch 08); it may repeat some of the above. ]

[ Editor: problem as stated there: ]
What is wrong with Humpty Dumpty's (and Smullyan's) reasoning in this
puzzle, and what is the correct probability?

Puzzle 61 in R. Smullyan's aptly named book _What is the Name of This
Book?_ is about Tweedledum and Tweedledee.  One of them always lies on
Monday, Tuesday, and Wednesday, and never on other days.  The other
always lies on Thursday, Friday, and Saturday, and never on other days.
Alice has a rattle that belongs to one of them and wants to return it.

        Several days later Alice again came upon just one of the
        brothers lying under a tree.  She asked the same question
        ["Who really owns this rattle?"], and the reply was "The
        owner of this rattle is telling the truth today."

        Alice pondered over this; she wondered just what were the
        chances that the speaker owned the rattle.

        "I know what you are thinking," said Humpty Dumpty, who happened
        to be standing nearby, "and the chances are exactly thirteen out
        of fourteen!"

        How did Humpty Dumpty ever arrive at those numbers?


No solution was found in the collected postings.

[ Editor: the question was posted by Col. G. L. Sicherman
<gls@hrcms.ATT.COM> in January 1994. ]