Riddle of the Three Prisoners: Who Knows Their Hat Color?

In summary: The emperor offers a possible solution, in which the person with the correct answer to his riddle will be set free. The first man in line is asked a question with no answer--he cannot see what colour the hat of the person in front of him is--and is thereby unable to answer. Next, the emperor has the second man put on a white hat and asks him the same question. The answer is still unknown to the first man. The emperor then has the third man put on a black hat and asks him the same question. Now, the first man knows that the hat of the person in front of him is black, and the second man knows that the hat
  • #1
drizzle
Gold Member
200
57
an emperor judged three prisoners to be executed, but he gave an exception to whomever gives the correct answer to his riddle will be set free. first, he asked them to line up, in a way where the first can’t see the two behind him, the second can see the one on front of him, and the third can see both [they choose their positions, the emperor didn’t interfere]. he then brought four hats, two black ones and two white, he puts a white hat on the first person’s head [remember the first person can’t see what’s on his head, but the other two behind him do], and puts another two hats on the others’ heads and he hides the forth hat [none of them see/know what color is that hat], now only one of them is 100% sure of his hat color, who’s that person and what color is his hat?
 
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  • #2
The only way one person can be 100% sure of his hat color is if he sees two hats of one color; he therefore knows he has the other color. The only person who can see two hats is the last man. Since we know the first man has a white hat, the second man must also have a white hat; ergo, the third man wears a black hat.
 
  • #3
negitron said:
The only way one person can be 100% sure of his hat color is if he sees two hats of one color; he therefore knows he has the other color. The only person who can see two hats is the last man. Since we know the first man has a white hat, the second man must also have a white hat; ergo, the third man wears a black hat.

hint: the emperor is an honest man!

that would be a direct help from the emperor to the third person, don't you think?
 
  • #4
If the first two have the same colored hats, the third instantly knows the color of his hat is not that color. If not, the second would be the only one to have certainty of knowledge of the color of his hat, because if the third person says nothing, this means that the colors must be different. Since the second man knows the color of the hat on the first, he then knows that his hat is black.

This is way too simple. :(
 
  • #5
didn't say it's hard, though it could’ve been if I didn’t mention the 100% sure thing, and of course without the hint :biggrin:
 
  • #6
Actually, the "100% sure" part is pretty presumptuous. If the first two people received white hats, and the 3rd prisoner was rather dumb, or if the 3rd prisoner wanted the 2nd prisoner to incorrectly guess that his hat was black, the 3rd prisoner could just sit there and wait for 5 minutes before saying anything, even though he has all the data necessary to guess his own hat color.

For that matter, we're assuming that the prisoners are "100% sure" that the emperor told them how many of each color hat there would be, and that he was telling the truth when he did so. The emperor could have lied and simply given them all white hats.

DaveE
 
  • #7
what if the third person sees 2nd guy wearing a black hat and the 1st a white.

edit nevermind.
 
  • #8
Ya. this riddle as stated is impossible. Nowhere in the riddle do you say that the prisoners are then expected to speak out loud, so that the other prisoners can hear, in real time if they know what hat they have. In fact you don't say anything about the prisoners being able to communicate or being able to see/hear if another prisoner is answering the emperor.
 
  • #9
ITS THE MIDDLE GUY, or the last guy

IF guy 1&2 were wearing hats of the same color guy 3 would know and say "i have the answer"
but if he dosn't - that because guy 1&2 wear hats of a different color, so the middle guy says "i have the answer" and knows his hat is the opposite color to the guy in front
 
  • #10
Last guy raises hand and says BLACK hat? lol
 
  • #11
The only person who is sure, it seems, is the Emperor.

The first man may have a white hat, but this allows that the second man has a black hat. Leaving the 3rd man with the possibility of having either color.

Or so it seems
 

Related to Riddle of the Three Prisoners: Who Knows Their Hat Color?

1. What is the Riddle of the Three Prisoners?

The Riddle of the Three Prisoners, also known as the Hats Riddle, is a classic logic puzzle that involves three prisoners wearing hats of different colors and trying to guess the color of their own hat. It is meant to test one's deductive reasoning skills and critical thinking abilities.

2. How does the Riddle of the Three Prisoners work?

The riddle goes as follows: Three prisoners, A, B, and C, are brought into a room and are told that each of them will be wearing a hat, either black or white. They cannot see their own hat, but can see the hats of the other two prisoners. The prisoners are not allowed to communicate with each other. They are then asked to guess the color of their own hat. If at least one prisoner guesses correctly, they will all be set free. If any of them guesses incorrectly, they will all be executed. How can they increase their chances of survival?

3. What is the solution to the Riddle of the Three Prisoners?

The solution to the riddle is for prisoner B to say the color of prisoner C's hat, and for prisoner C to say the color of prisoner A's hat. This way, at least one of them will guess correctly, and they will all be set free. If prisoner B sees that prisoner C is wearing a white hat, he will know that his own hat is black. If prisoner C sees that prisoner A is wearing a black hat, he will know that his own hat is white. Therefore, when it is their turn to guess, they will be able to make an accurate guess.

4. Are there any other solutions to the Riddle of the Three Prisoners?

Yes, there are other solutions to the riddle, but they all involve one of the prisoners making a guess based on the colors of the other two prisoners' hats. For example, prisoner A could guess the color of prisoner B's hat, and if he sees that prisoner B is wearing a black hat, he will know that his own hat is white. However, this solution relies on the assumption that there are only two possible hat colors.

5. What is the significance of the Riddle of the Three Prisoners?

The Riddle of the Three Prisoners is a popular puzzle used in logic and critical thinking courses to teach problem-solving skills and strategic thinking. It also has real-life applications, such as in game theory and decision-making processes. It highlights the importance of communication and cooperation in problem-solving situations.

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