-----Original Message-----
From: Large, Darien
Sent: Tuesday, January 05, 1999 1:39 PM
To: Sergio; Joseph; Ken; Bob; Michael; Alan; Large, Darien; Dustin; Ed;
James; Joe; M; Mark; Michael; Robert; Shawn; Travis; Elias; Chris
Subject: Of hats and seats and Kings: Another Puzzler from
Click 'n' Clack, the Tappet Brothers
Importance: High
OK, there's this King, see? And he wants to find the wisest man in the
Kingdom.
So, he calls the famous three wiseguys to his court and sits them in a
circle facing each other. (Since they didn't have chairs they all had to
sit on the floor; it's complicated why the Kingdom couldn't afford chairs
and it doesn't really matter to the puzzle so don't ask.) Then he says,
"In this bag I have some red hats and some white hats. I'm going to go
around and put either a red hat or a white hat on your heads, but I'm not
going to tell you what color it is."
This he now does. The three wiseguys can all see each other, but they
can't see the hats on their *own* head, of course. Now the king says,
"OK. If you can see a red hat, then raise your hand." All three wiseguys
raise their hands.
Then the king says, "Now, if you know what color hat you're wearing, I
want you to stand up. And by the way, you can all put your hands down
now."
Time passes. The King orders a pizza. The shadows lengthen (it was the
olden days). Finally one of the wiseguys stands up and says,
"I'm wearing a red hat." The King acclaims him as the wisest of all the
wise men in the Kingdom, and there was much rejoicing.
The question is, *how did the wisest guy know that he was wearing a red
hat?*
(There's a clue at the bottom of this message, but I still can't decide
whether it makes things more confusing or less confusing. I'd recommend
you only peek if you get absolutely nowhere--my feeling is if you make any
sort of progress on your own, then looking at the clue will only confuse
you. Weird, huh?)
As usual, I'd like to figure out how this puzzle goes to the more general
case-- four, five, six wiseguys and so on.
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Clue:
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The King actually puts a red hat on each of the wiseguy's heads; no-one is
wearing a white hat, but of course since each wiseguy can't see the hat on
their own head they don't know that.
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-----Original Message-----
From: Chris
Sent: Tuesday, January 05, 1999 1:55 PM
To: Large, Darien
Subject: RE: Of hats and seats and Kings: Another Puzzler from Click 'n'
Clack, the Tappet Brothers
Okay, the clue sucks. I was about to say, the guy who stands up sees a
white hat on one of the other two wisemen. In order for the other wiseman
he sees (the one with the red hat) to see a red hat then he, the guy who
stands up, would have to be wearing a red hat. (That's confusing).
However, if all three guys have red hats on, then I don't get it. Each of
the three guys could still think they had a white hat on.
The only thing that doesn't fit is the bit about shadows lengthening, but
I can't see that as being at issue.
Am I close?
-----Original Message-----
From: Large, Darien
Sent: Tuesday, January 05, 1999 1:58 PM
To: Chris
Subject: RE: Of hats and seats and Kings: Another Puzzler from
Click 'n' Clack, the Tappet Brothers
Importance: High
Oh, but the shadows do fit, they do they do they do. P.S. no, you suck.
On Wed, 6 Jan 1999, Angela wrote:
>
> P.S. We did realize that only two of the guys must be wearing red hats,
> but aren't sure how anyone would reach the conclusion that all three are
> wearing red hats (unless you're assuming the king is fair, because if
> one guy had a white had he wouldn't have the same chance at winning as
> the other guys).
I *really* like this bit of reasoning, which is totally correct logically.
I'm going to send it to the distribution list if that's OK once I get
home.
The Tappet brothers have a web page and actually explain the answer to
this puzzler (along with all their others). go to Puzzler Answer: The
Mad Hatter
I didn't actually understand it, though, until I talked it out with Keith.
Here's the easiest way I can explain it:
Imagine you're one of the wiseguys. You see at least one red hat, and the
other two wiseguys raise their hands along with you. This tells you that
there are *at least two red hats*. Since everyone can see everyone's
hands, you know that everyone else knows this as well.
So everyone knows that there are at least two red hats. Now you ask
yourself, "What color is my hat? Let's say I were wearing a white hat.
Then there are two red hats, which the other two wiseguys are wearing.
This means that each of them would only *see* one red hat each (they'd see
each other's red hat and my white hat).
"Now, since they each know there are at least two red hats (because of the
hands), but they can only *see* one red hat each, that means they'd
immeditately know the color of the hat they can't see; i.e., their own.
They'd both know they were wearing red hats, and they'd both stand up.
"But since they didn't stand up, that means they haven't figured this out.
Why would that be? Well, if they know there are at least two red hats and
they also *see* two red hats, they wouldn't be able to tell right away
what color their own hat is (because it could be white).
"But if they both see two red hats, that means that I must be wearing a
red hat as well! I'm wearing a red hat!"
In other words, the key to the puzzle here is that *time passes*. No one
can tell right away what color hat they're wearing, since everyone sees
two red hats.
To me it seems like a kind of "broken symmetry" type situation--the kind
of particle physicist problem that asks, "Since fundamental particles and
forces are symmetrical, how can they result in the kind of *asymmetrical*
configurations like galaxies that we actually see?" Surprisingly,
asymmetry *can* arise out of "nothing" though I don't know anything about
it.
The wisest guy figures it out only when he realizes that the situation is
symmetrical--when he looks around and sees that *no one* knows what color
hat they're wearing. If it's symmetrical then everyone must be wearing
the same color hat. The symmetry makes the problem *difficult*, but it
also makes it *solvable*. Once the wisest guy realizes that they're all
in the same boat, suddenly, he's able to jump ship and swim to shore,
leaving the others in the dust.
Yrs Dar
-----Original Message-----
From: Darien Large
Sent: Friday, January 08, 1999 11:22 PM
To: Virginia; Trey ; Tracy ; Todd ; Sheila ;
Sean Patrick ; Naomi ; Linda Mae ; Kevin ; Kevin
; Keith ; George ; Erin ; Dixie ; David
; David ; David ; Camille ; Blake ; Billy
; Billy ; Bblais; April ; Angela ; Andrew
Cc:
Subject: Tennis, anyone? (Was:Of hats and seats and Kings: Another
Puzzler from Click 'n' Clack, the Tappet Brothers)
Although the key to this puzzle involves the fact that *time passes* (the
wisest guy realizes that since no one knows the answer right away, that
must mean that everyone's wearing a red hat), it *also* depends on the
assumption that all three wiseguys are about as wise as each other.
Otherwise, the wisest one can't depend on the information he gets when the
time passes and no one stands up.
If the wisest guy who stood up were much wiser than the other two, it's
possible that those remaining, seated "wise"-guys hadn't even gotten as
far as realizing what the show of hands revealed. In other words, they
may not even have solved the first part of the problem before the king
tells them to stand up. In that case, the wisest guy can't depend on
their confusion arising from the fact that they both see two red hats;
they might be dumb enough not to stand up even if they see one red hat and
one white hat.
So what we're left with is that all three wiseguys have to be fairly wise,
*but not all equally wise*. Let's say that they're all exceedingly wise
(for those of you who don't know, "exceedingly wise" is the official last
level of wisdom before infinity)--they're about as wise as can be. Then
the time required to pass before the wisest one can deduce from the pause
what color hat he's wearing, can be made as small as you like--even small
enough that there's no *perceptible* passage of time. It would look like
one of the wiseguys stood up "right away".
What would this tell you if you were one of the wiseguys? If you saw one
of them stand up right away, I think you'd incorrectly deduce that you
were wearing a *white* hat--if one of the red-hat-wearing guys stands up
right away, but you can't tell right away what color hat you're wearing,
then that means you're wearing a white hat.
The problem with this, however, is that if you see two people with red
hats and you're wearing a white hat, they should *both* know right away
what color hat they're wearing. Since you're exceedingly wise, you know
this of course. So if you see two red hats but only one guy stands up
"right away" then you know that you're wearing a red hat--the wiser one
figured it out in that "imperceptible pause" and the other didn't. If
they both stand up right away, you know you're wearing a white hat.
Of course, this has the same problem that the original puzzle does. It
only reliably leads you to the answer if you can safely assume that
everyone is pretty wise to begin with--otherwise you're not sure what the
guy or guys standing up means. Again, though, I think it's pretty safe to
assume both that the three wiseguys are pretty wise and that they all know
it.
Now, imagine that not only are they all exceedingly wise, but they're also
*exactly* as wise as each other. This means they'd all stand up after an
exceedingly small passage of time, and they'd all stand up in unison.
What would this tell you if you were one of them?
First of all, it seems to me that the pause would be very short, because
you're all very wise. If you're all exceedingly wise, then you'll
anticipate this scenario even before the king tells you to stand up if you
know the color of your own hat. *The pause would be driven down to zero
and you'd all stand up right away.*
Uh-oh, now what? Again, put yourself in the shoes of one of the wiseguys.
The king tells you to stand up if you know the color of your hat, and all
three wiseguys stand up at the same time--with no pause. (It's a lot like
the hand-raising, isn't it?) At this point, it might occur to you, "Gosh.
Those two fellows, who're both wearing red hats, know what color hat
they're wearing. And they knew it right away, too. That must mean that I
was wrong--I'm actually wearing a white hat." Being good as well as wise,
you'd sit down again.
Of course, the other two wiseguys are thinking the exact same thing--and
they sit down right along with you. Then you think, "They thought they
knew what color hat they were wearing, but when everyone stood up at the
same time, they realized they were wrong, just like I did. This must mean
that they thought they were wearing red hats (which they are) but then
changed their minds for the same reason I did--they saw two red-hat guys
stand up and deduced (now incorrectly) that they were wearing a white hat.
Being good as well as wise, they sat down again."
It doesn't take very long for you to realize: "Wait a minute! If they saw
two red-hat guys stand up, then I must be wearing a red hat! I'd better
stand up again! I'm wearing a red hat!"
You can see where this train of thought leads from here, I think.
Of course, there's a problem with this scenario as well--the pause might
be very short, but it must be perceptible in order to convey the necessary
information. You have to notice that no one stood up right away, in order
to figure out that you're wearing a red hat. (If you didn't notice the
pause, you might just conclude that you're wearing a white hat, and the
two red hats figure it out "right away".) How long is long enough for an
exceedingly wise wiseguy? It'd be pretty short, that's clear. There's no
room for hesitation, because the other guys're liable to figure it out if
you wait *too* long. Again, it seems like the length of the pause will be
driven down to zero!
It occurs to me that you could try just *standing up* and see what the
other guys do. If they stand up after they see you stand up, what would
that mean? What about you standing up would tell them the color of their
own hats? If one of them sees you stand up right away, that should tell
him (incorrectly) that you see a white hat and a red hat. Now, since he
sees two red hats, he'd deduce (incorrectly) that he must be wearing a
white hat. He'd stand up.
Meanwhile the third wiseguy sees two redhat guys stand up--and he'd deduce
from this (incorrectly) that he's wearing a white hat. You think to
yourself (incorrectly), "Two redhats just stood up. This means I'm
wearing a white hat.
"But it was only my standing up that allowed them to realize what color
hat they're wearing, just as them standing up allowed me to deduce that
I'm wearing a white hat. How could they figure it out from that? Wait!
They could only do that if they saw I'm wearing a red hat, because a red
hat will only stand up if he sees a red hat and a white hat. Or, put
another way, a white hat will never stand up first (unless they're tricky
like I am--I didn't actually know the answer when I stood up, but they
don't know that). Now, if a red hat stands up when they see a red hat and
a white hat, they must both believe they're wearing white hats. Of
course, they can't both be wearing white hats, and in point of fact
they're both wearing red hats. If they both think they're wearing white
hats, that means they must both see two red hats. Which means I'm wearing
a red hat!"
Tennis, anyone?