***  Dottown Solution:  Part II  ***

Let's say that I have some secret, and you don't know it, but
our mutual friend J does.  If J goes to you and tells you my
secret, then you know my secret, but I don't know that you know.
If J comes back to me and apologizes for telling you my secret,
then you still know the secret, I know that you know, but you
don't know that I know that you know.  If J then goes back to
you and tells you about the apology, then you know the secret, I
know that you know, you know that I know that you know, but I
don't know that you know that I know that you know.

Let us adopt the notation {foo}_R to indicate R repetitions of
foo.  Thus, `{ quux}_4' expands to ` quux quux quux quux'.

If J keeps going back and forth between us making a total of N
statements, but you and I never discuss the matter, then,
for N >= 2, the statements

    You know{ that I know that you know}_A.
    I know{ that you know that I know}_B that you know.

hold for all A <= (N - 1)/2 and all B <= (N - 2)/2, but do not
hold for any A > (N - 1)/2 nor for any B > (N - 2)/2.

In other words, if J has made N statements (alternating between
us), then any sentence of the form ``... ... know that ... know''
(alternating between `I' and `you') is true for up to N
occurrences of the word `know', but false for all sentences with
more than N occurrences of the word `know'.

If you and I finally talk to one another and discuss this
matter, then we can say

    You know{ that I know that you know}_C.
    I know{ that you know that I know}_D that you know.

for any values of C and D, however arbitrarily large they may
be.

What does this have to do with Dottown?  Absolutely everything.

The stranger's statement in Dottown has the same effect as our
mutual discussion of my secret in the example above.

In the days before the stranger, if there were at least two red
dots, everyone in Dottown would know that there was at least one
red dot, but that doesn't mean that everyone would know that
everyone else knows that there was at least one red dot. 

Let's try to clarify this with a specific example.  Say JBP0 has
a blue dot, and JRP0 and JRP1 have the only two red dots.  All
three of them know that there's at least one red dot.  JRP0
looks at JRP1 and thinks,

    JRP0> ``JRP1 has a red dot, but that may be the only one, so
    JRP0> I don't know whether JRP1 knows that there is at least 
    JRP0> one red dot.  In fact, JRP1 knows that there is at
    JRP0> least one red dot if and only if I have a red dot.''

The blue-dotted JBP0 can look at JRP0 and JRP1 and think,

    JBP0> ``JRP0 and JRP1 both have the same color dot, so they
    JBP0> both have exactly the same information (mutatis
    JBP0> mutandis).  They both know that there's at least one
    JBP0> red dot, but I don't know whether JRP0 knows whether
    JBP0> JRP1 knows that there's at least one red dot.  In
    JBP0> fact, JRP0 knows that JRP1 knows that there's at least
    JBP0> one red dot if and only if I have a red dot.''

Now consider a more general case.

Let's call the residents JQP0, JQP1, ... , JQP[P-1], and assume
that there are R red dots.  Then, for any  a  from 0 to P - 1,
JQPa knows that there are at least (R - 1) red dots.  Also, for
any a and b, JQPa knows that JQPb knows that there are at least
(R - 2) red dots.  For any a, b, and c, JQPa knows that JQPb
knows that JQPc knows that there are at least (R - 3) red dots,
and so forth.  In general, for arbitrary a, b, c, ... , the
statement

    JQPa knows that JQPb knows that JQPc knows ... that there is
    at least one red dot.

holds for up to (R - 1) occurrences of the word `knows', but does
not hold for arbitrary a, b, c, ... for R or more occurrences of
the word `knows'.

Once the stranger announces to all that there is at least at one
red dot, everything changes.  After the announcement, the
statement

    JQPa knows that JQPb knows that JQPc knows ... that there is
    at least one red dot.

holds for any a, b, c, ..., and for any arbitrarily long
statement.  It doesn't matter how many `knows's there are; the
statement remains true.

The importance of the stranger's announcement, then, is the
information it conveys about what everyone else knows.  There
comes a point when everyone with a red dot will think, ``If I
had a blue dot, everyone else I see with a red dot would have
had enough information to ascertain their dot color and thus
would be dead; ergo I must have a red dot.''

If the stranger had gone around and whispered in each resident's
ear, ``There is at least one red dot,'' so that no one knew what
the stranger was saying to anyone else, then no one would die,
with the exception of a sole red-dotted resident.  (If only one
resident had a red dot, that person would die since the stranger
conveyed previously unknown information; no one else would die
because they don't know what the stranger said to the now
deceased person.)

In our Dottown, however, the stranger chose to make a public
pronouncement about the dots.

Punctus pronunctus est.  Alea jacta est.

(Latin for ``Them d00dz is d00md.'')