Results of

Charles'Friend's Deductive Powers

The Original Problem:

Maggie and Charles are at Zeno's late one night with friend Cathy, who is blind. A challenge is presented, involving the waiters, Steve and Bob. It seems that the waiters and Cathy are each given, unknown to them, hats that are either black or white, but that not all are wearing white hats, and they are told this. Steve can see the hats worm by Bob and Cathy, and Bob can see the hats worn by Steve and Cathy as well. Cathy, of course, sees nothing. Each is asked if they can figure out the color of their own hat from this information.

Steve answers no, as does Bob. Cathy thinks for a moment and exclaims that she knows the color of her hat!!

Question: What is Cathy's hat color, and how does she know??

Results Submitted:

Below is the result of your feedback form. It was submitted by Nora () on Monday, August 17, 1998 at 15:27:21

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username: ncorri@email.unc.edu

state: NC

city: Chapel Hill

country: USA

comments: Cathy's hat is white and she reasons as follows:

If either Steve or Bob saw two black hats, he would know that his own hat must be white. If either of them saw a black hat on Cathy's head and a white hat on the other's, he would know that his own hat must be white because if it were black, the other waiter would see two black hats and know his own hat was white. Therefore if Cathy were wearing a black hat, the waiters would both be able to deduce the color of their own hats; since they can't, she knows hers is white.

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Below is the result of your feedback form. It was submitted by Jason Johns () on Monday, August 17, 1998 at 21:29:41

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username: johnsj@moody.af.mil

state: GA

city: Valdosta

country: USA

comments: There are six possible combinations with black and white hats involving 3 people. The first 4 combinations which have Cathy either wearing a black or white hat and the waiters wearing different colored hats will yield an answer from one of the waiters each time. This is because one waiter will see 2 of a color and know that his hat is the opposite color. This leaves us with the other two possibilities. One is Cathy wearing black and the waiters wearing white and the other is Cathy wearing white and the waiters wearing black. In either senerio the waiters cannot tell which hat they are wearing because they both see one of each color and cannot tell which one they have. However, since Cathy is blind she cannot see which color hats the waiters have on and the only way she can tell her own color is to say, "the color of my hat is the exact opposite color of the two waiters hats". This is because it is having the same color hats that inhibits the waiters from figu!

ring out what hat they have on and, hence Cathy has on the odd colored hat.

--Good one Dr. Barnette

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Below is the result of your feedback form. It was submitted by PReay () on Sunday, August 23, 1998 at 23:33:06

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username: preay@email.msn.com

state: RI

city: Coventry

country: USA

comments: Regarding the hat problem:

Givens:

A. All hat's aren't white

B. Cathy hears both Steve's and Bob's answers before answering.

Assumptions:

A. If Steve or Bob looks at 2 white hats, he will recognize that his is black.

B. Bob hears Steve's answer before answering.

Solution:

A. There are only 8 possible hat combinations (2 hat colors, 3 hats). In the solution table below, let Steve represent the first column, Bob the second and Cathy the third. Also let B=Black and W=White for the hat colors.

S B C

(1) W-W-W

(2) B-W-W

(3) W-B-W

(4) W-W-B

(5) B-B-W

(6) B-W-B

(7) W-B-B

(8) B-B-B

B. Combination (1) is eliminated since all hats can't be white.

C. Combinations (2) & (3) are eliminated since in either case Steve or Bob would be looking at two white hats and recognize that his is Black.

D. Since Bob has heard Steve's answer, Bob knows that he and Cathy are not both wearing white hats. Therefore, if Cathy were wearing a white hat, Bob would know that his hat is black. Since Bob doesn't know the color of his hat, Cathy cannot be wearing a white hat.

Thus, solution (5) is eliminated and Cathy knows she is wearing a Black hat.

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Below is the result of your feedback form. It was submitted by James Holohan () on Thursday, August 27, 1998 at 05:48:12

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username: j.p.holohan@uclan.ac.uk

city: Preston

country: UK

comments: Steve answers no, so there must be at least one black hat on Bob or Cathy's heads. Bob then knows that Steve saw a black hat and therfore if Cathy's hat was white he would know that his hat was black. Therefore Cathy's hat must be black.

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Below is the result of your feedback form. It was submitted by Michael Patton () on Friday, August 28, 1998 at 23:41:05

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username: mfpatton@mindspring.com

state: al

city: montevallo

url: http://www.mindspring.com/~mfpatton

country: USA

comments: If either Steve or Bob sees a white hat, he knows that

he has a black hat on if the other doesn't know his hat color.

This is because if anyone sees 2 white hats, that person

instantly knows the color of his or her hat. Thus, Steve

and Bob can only be unsure if they see only black hats,

so Cathy has a black hat on.

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Below is the result of your feedback form. It was submitted by Avery Johnson () on Sunday, August 30, 1998 at 19:34:21

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username: johnal01@holmes.ipfw.edu

state: In

city: Fort Wayne

country: USA

comments: If Steve and Bob bothered to think before speaking (a good

assumption for any who work in so prestigious an institution

as Zeno's), Cathy can be sure her hat is black. A diagram

of all eight possibilities shows that the four in which she

has a white hat would allow either Steve or Bob to deduce

the color of his hat. This is an old problem, so there are

many discussions of it readily available. Making Cathy

blind, however, is a nice twist; it does not affect the

logic, but it shows that good sense is more important than

the mere five senses.

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Below is the result of your feedback form. It was submitted by () on Monday, August 31, 1998 at 10:11:53

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country: USA

comments: The Hat is black. All the hats are black.

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Below is the result of your feedback form. It was submitted by Mark Young () on Tuesday, September 1, 1998 at 10:34:20

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username: Mark.Young@AcadiaU.CA

state: NS

city: Wolfville

country: Canada

comments: There's at least one black hat. Steve knows this, so if he

saw two white hats, he'd know that his was black. He

doesn't know, so he sees one (possibly two) black hat(s).

Bob knows that there's a black hat, and he knows that Steve

sees a black hat (we'll assume he's able to figure out what

we did above). If Bob sees a white hat on Cathy, then the

black hat Steve sees must be on him (Steve can't see his

own hat), so he'd know his hat was black. He doesn't know,

so he must not see a white hat on Cathy.

Cathy knows Bob doesn't see a white hat on her (again we

assume she's bright enough to figure out what we did).

There are only white hats and black hats, so Cathy figures

her hat must be black.

Of course, the whole thing falls apart if Bob isn't bright

enough to figure out the ramifications of Steve's "no".

Let's try putting black hats on Steve and Bob and a white

hat on Cathy and see if Bob has what it takes....

...mark young

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Below is the result of your feedback form. It was submitted by Fred Weniger () on Tuesday, September 1, 1998 at 20:40:25

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username: fweniger@vcnet.com

state: CA

city: Simi Valley

country: USA

comments: Cathy's hat is black. If Bob & Cathy's hats were white, Steve would know his is black.

If Steve & Cathy's were white, Bob would know his was black. If Cathy's was white and

Steve could not tell, Bob would know his was black. Because Bob could not tell, Cathy's

hat must be black.

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Below is the result of your feedback form. It was submitted by Chris Horst () on Friday, September 4, 1998 at 13:39:47

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username: horstc@letu.edu

state: TX

city: Longview

country: USA

comments: Cathy's hat is black.

If Steve had seen two white hats, he would have known that his hat was black, since there could not be three white hats. Steve must have seen at least one black hat on Bob and Cathy's heads. Knowing this, Bob looked at Cathy's hat. If he had seen a white hat, he would have known that his hat was black. Therefore, Cathy knew that her hat was black.

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Below is the result of your feedback form. It was submitted by Paul Beach () on Sunday, September 6, 1998 at 14:00:10

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username: pbeach@gpu.srv.ualberta.ca

state: AB

city: Edmonton

country: Canada

comments: Cathy's hat is black. Since not all hats are white, if Bob saw the other two with white hats, he would know his was black. Likewise for Steve. However, since neither of these guys proclaimed they knew, Cathy realizes that both of them must have seen a black hat; HER black hat. There are problems, though. First, Cathy assumes that Steve and Bob have reasoned correctly. If either of them failed to reason properly, then poor Cathy will also fail. But, there is the possibility that BOTH Bob and Steve have black hats, and Cathy has a white hat. Steve would see Bob's hat, and then doubt his own, as would Bob.

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Below is the result of your feedback form. It was submitted by Maarten van Zuidam () on Tuesday, September 8, 1998 at 12:25:24

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username: m.l.vanzuidam@student.utwente.nl

state: OV

city: Enschede

url: www.angelfire.com/ne/zuid

country: Holland

comments: Steve doesn't know what color his hat is, that means that either

Bob or Cathy or both have a black hat. Because if Bob and Cathy

Would Both have white hats Steve would have known the color of

his hat was black (since not all three wear white hats)

Now it is Bobs turn. He knows he or Cathy (or both) must have

a black hat otherwise Steve would have known the color of his hat.

So when Bob would see that Cathy wears a white hat, he would know

his hat must be black. Since he doesn't know the color of

his hat Cathy most wear a black hat!

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Below is the result of your feedback form. It was submitted by Kyle Calderhead () on Tuesday, September 8, 1998 at 12:34:55

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username: chead@math.umn.edu

state: MN

city: Minneapolis

country: USA

comments: Black.

Steve can't see two white hats, so Bob or Cathy (or both) have a black hat.

Assuming Bob's on the ball, if he sees Cathy in a white hat, he could conclude he's got a black one after hearing what Steve says.

Since neither one knows, Cathy's got to have a black hat.

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Below is the result of your feedback form. It was submitted by Thomas Erlebach () on Wednesday, September 9, 1998 at 07:58:19

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username: erlebach@in.tum.de

state: BY

city: Munich

url: http://www.in.tum.de/~erlebach/homepage.html

country: Germany

comments: Cathy's hat is black! If it were white, either Steve or Bob

would have known the color of their own hat. To see this,

let's assume that Cathy's hat is white. If Bob's hat was

white as well, Steve would see two white hats and would know

that his own hat is black. But Steve didn't know the color

of his own hat, so he must have seen at least one black hat,

which means that Bob's hat is black. Bob can see that Cathy's

hat is white and, since Steve couldn't tell the color of

his own hat, Bob can figure out that his own hat must be

black. But Bob couldn't tell the color of his hat either,

hence the assumption that Cathy's hat is white has led

to a contradiction. That's why Cathy's hat is black and

she knows it. Of course, all of this assumes that Steve

and Bob are clever enough to tell the color of their hat

if they have sufficient information.

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Below is the result of your feedback form. It was submitted by Katrina Isaack () on Thursday, September 10, 1998 at 15:42:58

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username: isaackk@letu.edu

state: TX

city: Longview

country: USA

comments: Cathy's hat is black.

Here are the possible combination of hats (B for black

and W for white):

Steve B W B B B W W

Cathy B B W B W B W

Bob B B W W B W B

By considering these combos, if either Bob or Steve saw two

white hats, he would know his was black. Since neither of

them knew his hat color, all combos which involve Cathy and

one of them with white hats must be ruled out.

This leaves the following combos:

Steve B W B B W

Cathy B B B W B

Bob B B W B W

If Steve saw a white hat on Cathy and a black hat on Bob, he

could deduct his hat was black. Since he didn't this combo

could also be removed. The only combos left involve Cathy

having a black hat.

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Below is the result of your feedback form. It was submitted by James Doherty () on Monday, September 14, 1998 at 13:02:42

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username: mgbfred@aol.com

state: NJ

city: Haledon

country: USA

comments: Cathy's hat is white, the other two have black.

It says that not all are wearing white (therefore not all

black either). First, let's call Cathy 1, Steve 2, and

Bob 3. From both Steve and Bob's perception, if 1 and 2,

or 1 and 3 were both white, they could easily deduce their

hats to be black, since they cannot all be white.

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Below is the result of your feedback form. It was submitted by Rusty coan () on Monday, September 14, 1998 at 15:07:49

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username: rwcoan@valdosta.edu

state: Ga

city: Valdosta

country: USA

comments: cathy is wearing the black hat while steve and bob are wearing

white hats . when steve looks at bob and cathy he sees two

different colors and has no way of knowing what color his is

and the same goes for bob. while cathy thinks about this for

a few minutes she deducts that steve and bob have on the same

hats and hers is different

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Below is the result of your feedback form. It was submitted by () on Thursday, September 17, 1998 at 14:18:20

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country: USA

comments: She is wearing a black hat. The question say not all are wearing white hats. It did not say that could be wearing black hats.

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Below is the result of your feedback form. It was submitted by Ariel Steagall () on Saturday, September 19, 1998 at 00:55:53

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username: gingeu@freent.com

state: FL

city: Pensacola

country: USA

comments: Cathy's hat is white. Since she's blind, all she sees then is black. To, Cathy, Steve and Bob's hats are black making her hat white.

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Below is the result of your feedback form. It was submitted by Nattalie Ana Hollins () on Sunday, September 20, 1998 at 00:00:35

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username: morgaindre@aol.com

state: Wa

city: Roy

country: USA

comments: Her hat must be white, because....dang it!! I just had it a second ago....hmmm....Oh yeah. If Steve looks at the hats, and he saw both were white, then he would know that his hat was black. This is the same way with bob. There for, one of the two hats are white, and the only hat that both of them saw was cathy's.

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Below is the result of your feedback form. It was submitted by al gore () on Monday, September 21, 1998 at 16:41:58

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username: algore@o7.com

state: RI

city: Warwick

country: USA

comments: Cathy knows her hat is black.

Suppose Cathy's hat is white. Then if either Steve's or Bob's is also white, the other person would see 2 whites, and immediately know their own hat is black. So Steve would wait a moment - when Bob doesn't immediately say "black",

Steve would know his own hat was black. Same reasoning, of course, for Bob.

So Cathy waits two moments - if her hat is white, they will soon announce their own hat's color. Since they don't, hers must be black.

Cathy is assuming that the two waiters are smart enough to figure it all out. In real life, they'd probably still be trying to figure out what the question was, and thinking they'd better get a really big tip from these wierdos.

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Below is the result of your feedback form. It was submitted by david lowe () on Friday, September 25, 1998 at 18:11:47

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username: noodles74@mailexcite.com

state: ia

city: ames

country: USA

comments: Because not all the hats can be white, and the waiters see

one of each color hats on the others with out knowing there

own hat color, no sound decision can be made.

Steve and Bob see both black and white hats on the other two

players. They can therefore not answer.

Cathy understands this, and since not all are wearing white

hats, she knows that her hat is the odd color out of the

three. She knows that her color hat is opposite of both

steve's and bob's hat. not all white hats means that there

is at least one black one. Cathy knows this.

she's wearing black.

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Below is the result of your feedback form. It was submitted by Richard Deacon () on Sunday, September 27, 1998 at 09:35:15

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username: deacon@senet.com.au

state: SA

city: Adelaide

country: Australia

comments: Her hat is Black.

The waiters are told that not all the hats are white.

It is therefore implied that at least one hat is white

As they answer "no" it means that they must see hats of different colours.

If they saw 2 black hats they would know that theirs would be white and vice versa

Cathy is a smart girl :)

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Below is the result of your feedback form. It was submitted by T-Dog () on Monday, September 28, 1998 at 17:47:13

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state: GA

city: Valdosta

country: USA

comments: Cathy's Hat is white. The reason is Steve and Bob are both wearing Black hats and they can see her white hat and because not all are wearing white hats her hat has to be white.

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Below is the result of your feedback form. It was submitted by Austin Che () on Saturday, October 3, 1998 at 17:12:42

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username: austin.che@bigfoot.com

country: USA

comments: Black. If Bob and Cathy both had white hats, then Steve

would have known that he was wearing a black hat since all

three of them cannot be wearing white hats. Since Steve

answered no, Bob and Cathy cannot both be wearing white.

However, if Cathy was wearing a white hat, then Bob would

know that he was wearing a black hat, since Bob and Cathy

cannot both be wearing white. Therefore, Cathy cannot be

wearing a white hat, and so she is wearing a black hat.

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Below is the result of your feedback form. It was submitted by Terry Bowman () on Monday, October 12, 1998 at 22:58:06

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username: terry@century-times.org

state: OH

city: Columbus

url: http://century-times.org

country: USA

comments: Cathy's hat is white, because Steve and Bob's hats are both black. They each can see a black and a white, so they can't tell which color their own is. Cathy would deduce this from her knowledge that they don't know.

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Below is the result of your feedback form. It was submitted by Mathieu Weill () on Thursday, October 22, 1998 at 16:41:35

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username: matweill@aol.com

state: NY

city: New York

country: USA

comments: Cathy's hat is black. Because Steve says he does not know what color his hat is, Bob's and Cathy's cannot both be white (or Steve's would have to be black). Because Bob says he does not know, Cathy's hat cannot be white. If it were, his own hat would have to be black for Steve not to know what color his ow hat is. Therefore Cathy's hat must be black.

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Below is the result of your feedback form. It was submitted by Phillip Y () on Friday, October 23, 1998 at 14:18:06

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username: HeyPhil777@juno.com

state: ny

city: Beacon

country: USA

comments: From Cathy's point of view her hat is black, because

all she sees is black, she's blind therefore, colorless,

and dull, she sees nothing, except a shade. Say she takes

her hat off, and tries to look at it, she'll see black,

so that's what she thinks is the color of her hat, it all

depends on her point of view. She decides what it is, even

if it is white. Steve and Bob could tell her a million times

it's white, but she sees black, and what she sees, she

believes, sometimes a blind person can see what other people

can't.

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