Math Is Fun Forum

flip550

Member

Registered: 2011-07-27

Posts: 5

Issue with the black and white hat logic puzzle

Puzzle
Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten.

The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white.

Blindfolds are then placed over each man's eyes and a hat is placed on each man's head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.

The man in the rear who can see both of his friends' hats but not his own says, "I don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S hat says "I know!"

How did he know the color of his hat and what color was it?

Answer:
The man in front knew he was wearing a black hat because he knew the first man did not see two white hats and he knew that the second man did not see one white hat because if he saw a white hat, the second man would have known that his hat was black from hearing the first man's statement.

My issue:

What if the guy at the back see`s a black and white hat on the other 2, the guy in the middle can only see a white hat, if they both say `i don`t know` there is no way for sure for the man who cant see any hats to know what colour hat he is wearing, as there is 2 black and 1 white hats he cannot see.

Correct me if I`m wrong...

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Issue with the black and white hat logic puzzle

hi flip550,

These guys get information before they start = 3 black and 2 white hats;

and then get information from what they can see in front of them;

and finally get information from what they hear.

If the guy at the back could see two white hats then he would know his own must be black as there are only two white hats.

As he says he doesn't know, he must be seeing at least one, and maybe two, black hats.

So the guy in the middle hears this.  Either he himself, or the guy in front has a black hat on.

If he could see white ahead of him, he'd know he has the black.

But he also says he doesn't know.

So he must be seeing black.

The guy in front hears this and so he knows he has a black hat on.

Does that make sense to you?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

flip550

Member

Registered: 2011-07-27

Posts: 5

Re: Issue with the black and white hat logic puzzle

Hmm not really.. look at this picture
imageshack.us/photo/my-images/405/unled1hlq.jpg/

3 black, and 2 white hats total.

So man 1 is standing at the back, he can see the other 2 men, he can see one wearing a white hat(number 2 guy) and the other(number 3 guy) wearing a black hat.

So for man 1 there is no way to tell if he has a black or white hat as there is 1 white and 2 black hats that he cannot see. So he say's "i don't know"

Man 2 can only see man 3's black hat, which is irrelevant info to man 3 because he can only see one of the 5 hats.

There is no way for man 3 to say for sure "I know".

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Issue with the black and white hat logic puzzle

Hi flip550,

The man in the rear can know his hat only when the two on his front are W.

Since he doesnt know, the posibilities are { BW, WB, BB }.

The person in the middle can know his hat only when the one in the front is W.
But he doesn't know. So, what does that mean?

The possibilities are { WB, BB }.
That is, both end with a B. The person in the front can be sure he's wearing a Black hat.

Raymond smullyan is famous for such logic puzzles.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

flip550

Member

Registered: 2011-07-27

Posts: 5

Re: Issue with the black and white hat logic puzzle

The person in the middle can know his hat only when the one in the front is W.

this is wrong, he can only see one hat...
what if guy 2 is looking at a white hat, he would say "I know"?

Do i have to make a video?

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Issue with the black and white hat logic puzzle

what if guy 2 is looking at a white hat, he would say "I know"?

That's what I said.
If the previous person replied "I know", it is White, otherwise Black.

Do i have to make a video?

That's an overkill, but I think you may understand then!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Issue with the black and white hat logic puzzle

hi flip550

The man in the rear who can see both of his friends' hats but not his own says, "I don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S hat says "I know!"

It sounds like you are thinking that each man works independantly to the other two.  But that isn't what the problem says.  Maybe it needs a few extra words to make it clear that they comment in order; not all at once.

The back man makes a comment; the middle man hears it and then makes a comment; the front man now has both comments to take account of and that gives him the information he needs.

Don't waste time on a picture or a video because you cannot show what each man is thinking before he reaches his conclusion.

gAr and I, have both had a go at explaining this.  Please re-read what we have said. 

Bob

Last edited by Bob (2011-07-28 20:42:32)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

flip550

Member

Registered: 2011-07-27

Posts: 5

Re: Issue with the black and white hat logic puzzle

I have re-read all you replies, i still don't agree.

Ok hear me out one more time.

Man number 1 at the back can see everyone, he see's a black hat on the middle man, and a white on the front man, there is 2 black and 1 white left over that Man number 1 could possibly have on, so he says "i don't know". we all agree?

The middle man can only see a white hat on the guy in front of him. So he say's "i don't know".

The front man now from what he has heard say's "i know"?

And by the answer of this puzzle, it is meant to be black..... but it isn't, it's white.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Issue with the black and white hat logic puzzle

Oh boy!

Man number 1:  I agree with you.

So man number 2 knows that man number 1 cannot be seing two whites.

But he can see a white, so his own hat cannot be white.  So he wouldn't say "I don't know".  (Unless he was stupid!) 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Issue with the black and white hat logic puzzle

hi flip550

in your case the middle man would know that his hat is white.so that case doesn't match the propositions.

Last edited by anonimnystefy (2011-08-02 22:25:31)

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

anandbrar

Member

Registered: 2011-07-20

Posts: 70

Re: Issue with the black and white hat logic puzzle

Hi flip550

The middle man can only see a white hat on the guy in front of him. So he say's "i don't know".

Well suppose that all the men are super-smart and are trying to help each other by making logical answers.

So, now the middle man, when he's sees that the hat in front is white, he won't say 'i don't know' rather he will say 'i know' according to the following reasoning:

1) His hat could be black/white.
2) If it was white(and since the man in front also has white hat) then first man should have said 'i know'
3) Since first man doesn't say 'i know', hence his hat is not white.
4) Hence his hat is black..."He Knows!"

Simple enough?

Sorry if my english is below par.

---

"widen your gaze... extend beyond the obvious..."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Issue with the black and white hat logic puzzle

hi anandbrar

Sorry if my english is below par.

No, it's not.  Your English is excellent.  I'd believe you if you said you were born in Buckinghamshire, UK. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anandbrar

Member

Registered: 2011-07-20

Posts: 70

Re: Issue with the black and white hat logic puzzle

Thanks Bob
:D

---

"widen your gaze... extend beyond the obvious..."

flip550

Member

Registered: 2011-07-27

Posts: 5

Re: Issue with the black and white hat logic puzzle

Hi flip550

The middle man can only see a white hat on the guy in front of him. So he say's "i don't know".

Well suppose that all the men are super-smart and are trying to help each other by making logical answers.

So, now the middle man, when he's sees that the hat in front is white, he won't say 'i don't know' rather he will say 'i know' according to the following reasoning:

1) His hat could be black/white.
2) If it was white(and since the man in front also has white hat) then first man should have said 'i know'
3) Since first man doesn't say 'i know', hence his hat is not white.
4) Hence his hat is black..."He Knows!"

Simple enough?

Sorry if my english is below par.

This!

it finally makes sense gaahh sorry for wasting your times

sandesh234

Member

Registered: 2011-08-17

Posts: 7

Re: Issue with the black and white hat logic puzzle

Your not considering the scenariio when middle peson and front man both have worn black hats.
Last person sees 2 blacks and yet unaware of his colour. He says i dont know. middle person can not be sure either
Middle     Front   
w            W                SOLVED BY LAST PERSON
W            B            CAN ONLY BE SOLVED BY FIRST PERSON
B            B            CAN ONLY BE SOLVED BY FIRST PERSON
B            W            Middle man solves