***  Dottown hints  ***

Below are a few fundamental observations about the Dottown
puzzle.  I have called them lemmas, the term used in
mathematics for small theorems that lead to the proof of a
major theorem.  Of course, I won't bother to prove them since
they are `intuitively obvious.'  :-)

Remember that R is the total number of persons with red dots,
B is the total number of persons with blue dots, and P is the
total number of persons, so P = R + B.

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Lemma 1.  All persons with the same color dot have exactly the
same information, so if they die, they will all die the same
night.

   Each person with a red dot will see exactly the same number
   of persons with red dots and blue dots that every other
   person with a red dot sees.

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Lemma 2.  If anyone dies, then all persons in Dottown will die
on the same night, or on consecutive nights.

   If all ascertain their dot colors on the same day, they will
   all die that night.  If, say, all those with red dots die one
   night, those still living (if any) will notice the next day
   at noon that there are no red dots left, thus they must have
   blue dots, and will therefore die that night.

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Lemma 3.  Consider one resident of Dottown, called JQP.  Let R_v
(for "Red dots visible") be the number of red dots that JQP sees;
let B_v be the number of blue dots that JQP sees.  Then
R_v + B_v + 1 = P.  Consider the following statements:

   1)  JQP_has_red     (JQP has a red dot)
   2)  R = R_v + 1     (there's one red dot that JQP doesn't see)
   3)  B = B_v         (JQP sees all the blue dots)

   4)  JQP_has_blue    (JQP has a blue dot)
   5)  R = R_v         (JQP sees all the red dots)
   6)  B = B_v + 1     (there's one blue dot that JQP doesn't see)

Then Statements 1, 2, and 3 are equivalent;
     Statements 4, 5, and 6 are equivalent;
     Statements {1, 2, 3} are the logical opposite of {4, 5, 6}.

Thus, either (a) 1, 2, and 3 hold, with 4, 5, and 6 false,
          or (b) 4, 5, and 6 hold, with 1, 2, and 3 false.

So, if we know the true/false value of any of the six
statements, we know the value of all of the statements.

If JQP knows the value of any of the six statements, JQP will
die that night.

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Hint:

The stranger said that there is at least one person with a red
dot.  What happens if the stranger was lying?

That is, everyone has a blue dot, no one has a red dot, but
everyone believes the stranger anyway.  What happens?