On a tiny remote island lives a primitive tribe of philosophers, 
whose members all have a peculiar birthmark -- on
their forehead is a dot which is either red or blue.  The color of
an individual's dot has assumed great social importance, and indeed the
ultimate quest of any individual's life is to determine the color of their
own dot.  If anyone succeeds in doing so, that person has attained 
enlightenment, and with no reason left to live  commits ritual suicide that
very midnight (by doing a swan dive into the island's active volcano while 
yodeling, but the method is unimportant).

Of course, they don't have any mirrors, and there is a strict tabu
against anybody giving information about other people's foreheads;
consequently, nobody can actually determine the color of their own dot.
Island life is very stable, everybody sitting around all day philosophizing 
and thinking; they have all become very accomplished logicians.

One balmy afternoon, an explorer from the outside world finds the island, 
and innocuously mentions "I see each of you have either a red dot or a blue 
dot on your forehead; how fascinating!"

Q: What happens?

Hint: Consider the explorer saying something even weaker: 
"I see at least one of you has a red dot."

(If you want to reply to this to discuss, please label your subject-line
with 'Spoiler' or 'Possible Spoiler?' or something like that. 
No fair if you've just heard the answer before from somewhere else!
I'll post some mild hints after the break, or sooner if there is demand.)




[spoiler follows, don't peek 'til you know]

A: They realize they will all be dead within k+1 days, where k is the
number of people with red dots.
They are understandably distressed -- some current tribe members
think that enlightenment is overrated -- and they consequently
toss the explorer into the volcano that very evening.






Real Q: If there were two or more people with red dots, the explorer gave
no information that wasn't already known.  So why did that change anything?


A: In an island with exactly 2 red dots, yes
everybody knew there was at least one red dot, 
not everybody knew that everybody knew there was at least one red dot.
(In particular, each of those two red-dots wasn't sure whether the other
red-dot knew there was a red dot on the island.)

In an island with exactly 3 red dots, it's the same but once removed:
while everybody knew that:
  everybody knew that there was at least one red dot,
not everybody knew that:
  everybody knew that everybody knew that there was at least one red dot.

One way to interpret the reasoning is by an inductive proof,
proving that "if nobody has died in k days, then there are at least
k+1 people with red dots.", and the explorer provided the base case.





<comment>
Minor assumptions being made --

It is assumed that everybody
                                              believes the explorer, and
                     believes that the others believe  the explorer, and
believes that others believe  that the others believe  the explorer, etc.

Also, it is assumed
that no red-dot otherwise dies of natural causes before day k.
(Hmm, perhaps the tribal elder has to make a tough choice after the
explorer's leak, to sacrifice one red-dot, for the good of the community.
Or, would the elder's choice tip away as much (or more) info than the explorer?
(I don't think so.)
</comment>