```Author: docbill (Dr. Bill C. Riemers)
Date: 2004-Jan-17 17:37:22

Three men are discussing who is the smartest.  A woman tired of hearing the
boasting, decides to try an intelligence test.

She blindfolds all three, and places them sitting in a circle facing each
other.  She then paints a dot on the top of each one of thier heads.
Finally she explains the rules.

When she counts to three all men will remove their blindfolds at the same
time. The dot on their own head is either white or black.  If they see a
white dot on either of the other two men they should raise their hands
without delay.  The first person who correctly identifies the color of the
dot on his own head and can state why is the smartest.

Before she can finish counting, Bob, the smartest man in the room correctly
states the color of the dot on his head, and his reasoning.

What was the color, and how did Bob come to his conclusion?

Bill

Author: flint ()
Date: 2004-Jan-17 18:26:09

mmm.. I think I have it this time. Let's enumerate the possibilites:

Dot sequence    men_who_raise_hands

w,w,w   1,2,3
w,w,b   1,2,3
w,b,w   1,2,3
b,w,w   1,2,3

w,b,b   2,3
b,w,b   1,3
b,b,w   1,2

b,b,b   no-one raises hand

When there are atleast two whites, atleast one of the men cannot be sure of
his color (if all the three are white dots then none of the men can
determine his color). Assuming the woman would consider being fair to all
of them, she does not color more than two men in white.

When exactly two of the colors are white, then the two men who see a black
and a white will raise their hands. However, either one can deftly speak out
the answer first (as black) before actually raising the hand, which again
would be unfair for the man who sees both black and is depending on one or
the other of the men to determine the color on his own head.

This leaves the woman, in the interset of fairness, with the only option of
coloring all three black.

And thus, Bob could have answered before the count was over!

(I hope a neophyte's inanities are forgiven on bboard. ":>)

-F

Author: docbill (Dr. Bill C. Riemers)
Date: 2004-Jan-17 19:26:48

You are getting close.  Care to try again?

Bill

Author: migo (Mikhael Goikhman)
Date: 2004-Jan-17 19:39:11

I don't think I agree with your conclusion. :)

It seems probable that the answer is symmetric, so noone benefits.

Bob may think, if all colors are black, then noone raises the hand and
all three know they have black points. So it is unlikely all three have
black points. It is much more likely that all three have white points.
So noone may immediately determine his color, but after some thoughts the
smartest would determine his white color (because all others are silent).
I would say that all colors are white.

However we should not forget that the woman logic is in no way something
deterministic, so any other answer may be correct as well. There is no
enough information about the woman's mentality and intents. :)

BTW, flint, you may depost your first 2 messages if you want.
Press 'd 1.1' twice.

Author: flint ()
Date: 2004-Jan-17 20:05:06

Thanks migo for the help.

Thinking about the puzzle, what is most surprising is the fact that Bob
answers before the woman has finished counting. Hence his reasoning is
entirely based on his pre-evaluated conjectures of the number of raised
hands, which of cousre are inevitably based on assumptions about a woman's
intellect and morality.

While my sexist proclivity wanes in the pursuit of an intellectual train of
thought, I should think that Bob was the smartest because he had bought out
the answer well in time from the woman, possibly with an amorous foreplay
leading to the pinnacle of .... ":>

Am I going the right way, Bill?

-F

Author: docbill (Dr. Bill C. Riemers)
Date: 2004-Jan-18 00:31:22

Migo has it.  Here is how it played out...
........

Jane counted, "One, two, ..."

Before Jane could reach three, Bob interrupted, "I know the answer."

Jane queried, "Perhaps you do not remember the rules?  You need to know
both the color of your dot, and provide a conclusive argument how you know
it."

"Well Jane," Bob began smuggly, "my dot is white.  The reason is quite
elementary.  I only needed to know the rules, and that this was an
intelligence to determine which of us is smartest.  None of us would have
agreed to this test if we doubted your abilities to prepare a worthy test.
Hense I must conclude you painted my dot an all others the one color that
would provide an intelligence test.

"If you just wanted to be fair, you could have just given us all black
dots.  But then nobody would raise their hands.  Anyone who didn't realize
that ment we all had black dots would be an idiot.  Ergo, that would not
have been a intelligence test, but just a race to determine who could blurt

"If instead you had given just one of us a white dot, the two of us with
the black dots would raise our hands.  The first to realize the person with
the white dot was not going to raise his hand, would of course know the color
of his own dot.  That person would of course be the one with the white dot,
since who else could know sooner that he did not intend to raise his hand.
So again, this would not be an intelligence test, but the winner would be
determined by who you placed the white dot on.

"If instead you had given just one of us a black dot, everyone would have
raised his hand.  Either of the two people seeing a white dot would know
the fact they raised their hands ment their own dot was black.  Ergo, this
would be a race between the two with black dots, and the man with the white
dot could not prove his intelligence.

"The only option left is to give us all white dots.  In that case all of
use would raise our hands.  Only after one of us realized that the other
two not blurting out the answer means his own dot must be black.  This this
was the only true option that could determine who was the smartest.

"Having all reasoned this in advance, I knew I had to have a white dot on

Bob removes his blindfold and stares in disbelief at the two black dots on

Everyone starts to laugh, except Bob.

Jane chuckles, "Yes Bob, you are the smartest of all the men here, but you
are also a fool.  For only a fool draws a conclusion in the absense of
endlessly.  So I have proven once and for all that you are indeed the
smartest of all the men in the room, but not the smartest person in the
room."

:)

Bill

Author: migo (Mikhael Goikhman)
Date: 2004-Jan-18 01:14:32

Yes, this is a nice story proving that women are the smartest among us in
the sense of practicality. Are they also the smartest in the sense of pure
(i.e. theoretical) intellect? No way, girls.

However, Bill, did you copy and paste? You made 4 mistakes in 2 sentences
where you call "black" color "white", and "white" color "black". I.e.:

Either of the two people seeing a white dot would know the fact they
raised their hands ment their own dot was black.  Ergo, this would be a
race between the two with black dots, and the man with the white dot
could not prove his intelligence.

should be:

Either of the two people seeing a black dot would know the fact others
raised their hands ment their own dot was white.  Ergo, this would be a
race between the two with white dots, and the man with the black dot
could not prove his intelligence.

Author: docbill (Dr. Bill C. Riemers)
Date: 2004-Jan-18 01:48:37

You got me there.  I typed in the original problem wrong, as it was suppose
to be they raise their hands if they see a black dot.  So I had to reverse
the answer.   Obviously I missed a few spots.

Bill
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