The Hats

Logic Level 3

There are 2 are black hats and 3 white hats. Three of them are selected at random and placed on three people standing in a straight line. The person in the back speaks first.

What color is the person in the front wearing?

Note: Assume all of them are perfectly rational, and each person can only see the hats of the people in front of them.

Black White

14 solutions

Kimi P
Feb 6, 2016

If the front two people were wearing black, then the 3rd person would know he was wearing white. But since he didn't know, we can assume they both weren't wearing black. If the 1st person was wearing black, then the 2nd person would have known he was wearing white, because if he was wearing black then the 3rd person would have known what he was wearing. Since neither of them knew, the first person can assume that he wasn't wearing black.

But this solution requires the second person to know that the third person cannot tell what Third is wearing. The question just stated that Second and Third don't know what they themselves are wearing, not that they know about each other's knowledge

Sam Needham - 5 years, 4 months ago

Yes, that's correct. I've updated the problem statement.

Eli Ross Staff - 5 years, 4 months ago

If they are all perfect logicians then they would be confused by the word THEN because it implies that mr middle is awaiting information from mr back. If mr fronts hat were white then mr middle would not need information from mr back, so the implied wait suggests mr fronts hat is black and that mr middle would know his hat colour once mr back had spoken. It can't be the case that mr middle is both waiting for mr back to speak and doesn't know his own hat colour once mr back has spoken. So they are either not all perfect logicians or the question is mis-stated and ambiguous. I'd remove the THEN, when it is then clear that the front persons hat is white - once mr middle and mr back see a white hat on mr front they can exclaim (simultaneously or one after the other - but without an implied conditional dependency) that neither of them know their hat colour.

Andy Gregory - 4 years, 9 months ago

Bullshit. The problem clearly states that each person can only see the color of the hat of the person right in front of them. But Kimi P said, quote : "" If the front two people were wearing black, then the third person would know he was wearing white"". But this is wrong, because this assumes that the third person could see the color of the two in front of him. But as I just pointed out, he can only see one !!

If it was said that the third person could see the two in front, this problem would have belonged in level 1 !!

DarkMind S. - 4 years, 9 months ago

For a second I thought the same thing. But actually, the problem says the each person can see the hats of the PEOPLE in front of him, implying that the third person can see the hats of the first and second person.

David Ortiz - 3 years, 9 months ago

Nicely done. When I was a kid, I remember reading a problem in a book just like this, about a monster who kidnaps three people, puts the hats on them, and tells them they figure out their color or risk being eaten (if I remember that correctly). Ahh, memories! For the sake of being thorough, though, the problem probably should explicitly include the assumption that each of the men know there are exactly 2 black hats and 3 white hats.

David Ortiz - 3 years, 9 months ago

Altho the back guy would know IF the 2 in front were both wearing black but you did Not consider if the front guy was wearing White & the middle guy was wearing white or black THEN the back guy still does not know if he is the other white or black & neither does the middle guy . the question is a FAIL .

Mary Newton - 2 years, 7 months ago

The question ONLY asks about the FRONT guy. The fact that the back guys doesn't know just means that the front two are not both black hats. And the fact that the middle guy doesn't know means the front guy is not a black hat, otherwise the middle guy would figure that he had a white hat since the possibility of the first two being black hats was eliminated. Thus the front guy, though he does not know what the others are wearing, knows that he is wearing a white hat, which is what the question asked.

David Ortiz - 2 years, 7 months ago

Guillermo Templado
Feb 13, 2016

The last one in the line don't know the color of his hat, so the other 2 cannot both wear black. It must be either both white, white - black or black-white.

The middle one in the line can see the front one's hat. If the front wears black, then the middle must know his hat color(white) but he doesnt. Therefore, the front wear white

I said white and was counted wrong.

Robert Merrell - 3 years ago

Jonathan Dapadap
Jun 4, 2017

It's nice! This is my understanding.

• It is given that three hats are selected at random from 2 black and 3 white hats.

• If the two person in front of the person in the back is both wearing black hats, the person in the back will know that he is wearing white, since there is only two black hats. Otherwise, Black and White or White and White, the person in the back won't know his.

• The person in the back said, "I don't know what color my hat is!"

• Since the person in the back don't know, it is either one is wearing white and the other is wearing black (BW/WB) or both in front are wearing white hats(WW).

• If the person in front is wearing black, the person in the middle will know that he is wearing white (BW).

• If the person in front is wearing white, the person in the middle won't know his since he is either wearing black or white (WW or WB).

• After the person in the back made his statement, the person in the middle said,"Neither do I!"

• Since it is given that they are all rational, we can conclude that the person in front is wearing white hat.

• Therefore, the person in front is wearing white hat.

by: Jonathan Saut Dapadap... wew

Vishwa Soni
Feb 21, 2016

Simply, the probability that the front person will wear a white hat is 3/5 and that he will wear a black hat is 2/5. So, we can assume that the person will wear a white hat.

This is a wrong solution. The question is made in such a way, that there's a definitive answer, not a probability. Say we have a biased coin, being 60 % for heads and 40 % for tails. You cannot say that it will land heads every time just because the probability is higher.

Nicolai Kofoed - 4 years, 9 months ago

Here is another way to look at it. Below are all the possibilities of hats:

1) w w w

2) b w w

3) w b w

4) b b w

5) w w b

6) b w b

7) w b b

Now, notice that the #7 would let the person in the back know he was wearing a white hat since the other two were wearing black, so it has to be one of the top six combinations. Now notice something else, if the third person is wearing a black hat, the only solutions left are with the second wearing a white hat, so #5 and #6 are also out. That leaves us with the first four, in which the first person is wearing either black or white, the second is also wearing black or white, but the third person can only be wearing white.

Ricardo Almeida - 4 years, 3 months ago

Why #5 and #6 are out??

Felipe Maion - 3 years, 11 months ago

Felipe, #5 and #6 are out because the second person also does not know what color his hat is. Remember that they are all perfectly rational, so when person 1 says he doesn't know his own hat color person 2 has that information to work with and can discount #7. Looking at possibilities #1 through #6, the only way that person 2 can conclude he doesn't know his own hat color is by looking at a white hat, which only occurs in #1 through #4. Knowing that #7 is out, if person 2 saw a black hat he would conclude his own hat is white.

Ricardo Almeida - 3 years, 10 months ago

Suppose the following things were said by the first and second guy respectively: "I don't know the color of my hat." "I know mine, though."

Then the guy in front would be wearing a black hat, regardless of hat probability. See, sometimes a probabilty approach is not the way to high fashion.

Fabricio Kolberg - 2 years, 12 months ago

Carl Moore
May 10, 2018

I came up with same thing. From & middle persons can't both have black hats, because the rear person would then know he must have white hat. If front person has black hat, middle person would know he himself has white hat, because front & middle persons don't both have black hats. But middle person doesn't know his own hat color, so we conclude that front person has white hat.

By the way, I saw "are 2 are" in this puzzle; 2nd "are" should have been omitted.

Brian Coari
Dec 18, 2016

"There are 2 are black hats" was a rough start, but fun puzzle, keep at it!

There has to be at least one white hat on the first two people or else the person in back would know both black hats had been used and would therefore know his or her hat would have to be white.

The person in the middle would know that one hat among the first two would have to be white, so the person in the front must have a white hat otherwise the person in the middle would know his or her hat would have to be white.

John Mckenna
Nov 25, 2018

No solution possible. It is not given that the second and third person spoke. Maybe they thought those statements to themselves, and said nothing, if so, no solution. a BRILLIANT puzzle? hardly.

Charles Turner
Sep 10, 2018

I wish whoever sets these questions would get the English Grammar correct. The person in the middle's statement is ambiguous, it could mean he doesn't know the colour of his own hat OR he doesn't know the colour of the man at the back's hat.

Fabricio Kolberg
Jun 17, 2018

The first guy , a smart and elegant fellow, cannot figure out the color of his hat , what a bummer. If he had seen two black hats, then he'd know that he's wearing a white one , as no black hats would remain unclaimed. However, his friends' hat colors aren't helping him, so he saw at least one white hat among the guys in front .

From the first guy's answer, the second guy , also a smart and elegant fellow, cannot determine the color of his own hat . All he knows, from his buddy's statement, is that the first guy saw at least one white hat . If the second guy had seen a black hat, then he'd know the white hat is his own . But alas, he sees a white hat on the third guy , so he cannot guess.

Long story short, the dapper and clever gentleman in front wears a white hat , and wears it with style!

Three young men in hats, so elegant! But alas, as it is the fate of all characters in logic puzzles involving hats, they have to guess the colors of their hats, put on their heads by a mysterious unseen force. At least their lives are not at stake this time, right? Maybe that's why they're smiling so widely, this is a game for them.

Mike Holden
May 30, 2018

The back man deduces that he will only know the colour of his hat if the two in front are black. But he doesn't know.

Knowing this, the middle man deduces that he will only know the colour of his hat if the one in front is black. But he doesn't know.

Knowing this, we can deduce that the front man's hat is white.

Masa Mune
May 11, 2018

If the person in the back claims he doesn't know the color of his hat, the only way for the middle guy to not know the color of his hat is if the first guys hat is white.

B-B-W = Only if the guy in the back is one of those special people could he not know the color of his hat.
B-W-B or B-W-W = if the guy in the back does not know the color of his hat, then the middle guy knows his hat is white because the guy in front has a black hat

W-W-W or W-W-B or W-B-W or W-B-B = are all possibilities. So if the first hat is white, then it doesn't matter if the middle guy is wearing a black or white hat. The guy in the back cannot know the color of his hat. And because there is a 50/50 chance that his hat is either color, the middle guy doesn't know what color his is.

Rohit Kulkarni
Aug 28, 2017

The first 2 guys have 1 B and 1 W hat amongst themselves. Since the second guy sees a W hat, he says "Neither do I".

Gabi Mercea
Jan 30, 2017

I have no idea how this got here. It clearly could be either black or white, based on probability.

There is a chance all hats are black
2 black and 1 white
1 black and 2 white

Based on probability and the rationale of the people, for the first situation most people will say it's white, as there is a very low chance the remaining hat to be black.

For the remaining situations, the same thinking applies but the person has a higher chance of success.

You cannot blatantly say it's white, as it's not entirely correct.

Arjun SivaÞrasadam
Feb 13, 2016

Nice Problem !!!

The Hats

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