The Most Intelligent Prince

A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.

The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room.

The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed.

You are one of the princes. You see 2 white hats on the other prince's heads. After some time you realize that the other prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat?

Note: You know that your competitors are very intelligent and want nothing more than to marry the princess. You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery

• This is a fairly known and easily google-able question. Sep 6 '16 at 13:15
• I didnt know, My dad just emailed it to me Sep 6 '16 at 13:16
• Pretty sure, this will be marked as a duplicate soon...
– Sid
Sep 6 '16 at 13:16
• en.wikipedia.org/wiki/Prisoners_and_hats_puzzle Sep 6 '16 at 13:17
• @randal'thor which actually makes it a lot easier to solve...
– Sid
Sep 6 '16 at 13:21

Let's call the princes A, B, and C, with you being A.

• If you are wearing a black hat, then B will see one black hat and one white hat.

• B knows that if he is wearing black, then C can see two black hats (A and B), so C will know his own hat is white since there are only two black hats altogether.

• Since C doesn't speak, B will eventually realise that he must be wearing white and say so.

• Since B doesn't speak, the above assumption was wrong and you must be wearing a white hat.

• Note that the whole premise of the question is that, while the two other princes are intelligent, they are less intelligent than (or slower) you. Since everything is symmetric here, they otherwise could apply the exact same reasoning and answer before you... Sep 6 '16 at 13:19
• @ClementC. Sure. But that doesn't change any of the logic here, right? Sep 6 '16 at 13:22
• Nope, just a strange thing. The king ends up ranking sharpness/speed instead of intelligence. Also, another answer could be that the king is either a cheat, or fair. If he's fair, then all princes must have the same chance of marrying the princess. For that to happen, all their hats should be the same color, otherwise it's definitely unfair. (But that's beyond the point of the original question, I guess -- this is not tagged lateral-thinking.) Sep 6 '16 at 13:24
• Mightn't a truly intelligent prince realize that others may deduce false information from silence; leading to their deaths, the weakening of their kingdoms, and the strengthening of the prince's nation's positions? Sep 6 '16 at 13:29
• @Sconibulus Now you're venturing into the realm of lateral thinking ... Sep 6 '16 at 13:30

This could be it

White

because

If you were wearing a Black hat, then both the other prince would be seeing a white and a black hat in front of them and after some time, and after noticing that the other one (also with white) couldn't tell the correct hat, one of them would have spoken that he has a white. But since you also are wearing a white hat, none of them can say for sure what is their color

• I know I am a bit late but the captcha thing took a hell lot of time Sep 6 '16 at 13:22
• Captchas are horrible :-( Sep 6 '16 at 13:23

For it to be a fair test it must be symmetric, and thus it is unnecessary to see the other princes' hats: the only way it can be symmetric is if all three of us have the same colour hat, so we must all have white hats.

note that in the case with 2 black hats and one white hat, the prince wearing white hat can infer that he is wearing white hat, and instantly win by saying that. however, he risks nothing by withholding his answer. his silence can be interpreted by 2 other princes as seeing one black and one white hat and therefore not being able to infer his hat's colour. this would lead them to believe they have a white hat. so the white hat prince can just wait for the other prince to give a wrong answer getting him killed, and only then claim victory. a truly intelligent prince would consider this scenario, so the whole deduction in https://puzzling.stackexchange.com/a/42134/45528 fails at step 1.

this makes me believe the correct answer is to deduce that

all princes must have white hats for this to be a fair test.

• Hi, and welcome to the Puzzling SE! This is a very good observation, but unfortunately it doesn't answer the question, so it doesn't really belong in an answer. Given that you don't yet have the "post comments" privilege, that's quite understandable. You can gain some initial rep by answering questions, or posting puzzles. Be sure to take the tour too!
– Bass
Feb 17 '18 at 19:04
• the last sentence contains the answer :) i should probably put it in a spoiler Feb 19 '18 at 9:59

Rand al'Thor's answer seems to be right.. So is Clement C's observation..

Consider this scenario... since the answers are being based on "the unwillingness of the competitors to answer"...

When saying prince A has White, just because B & C are unable/unwilling to answer, you are assuming that A is more intelligent/faster. You have forgotten that even A has hesitated to answer so far, yet B & C were unable to answer. So we are safe to assume and consider that when A has a black hat, B & C (a bit slower as assumed earlier) would still be hesitant to answer. So why couldn't A's hat be Black?

It's never said that the test is fair for all three, it's only a fair test of intelligence.

I'd rather play a trick... I'd ask the two princes to reveal my hat's color and the first to answer can take the princess in exchange (she's mine to give since I am the first to deduce the color of my own hat) since I know 'my competitors want nothing more than to marry the princess'... and even they lied to me, I might end up dying and then they would ever be able to deduce their own hat, so they wouldn't lie...

Since the king is a man of his word (and I never declared that I am), I would get the princess since I deduced the color right.

I don't see any other way it could be a fair test of intelligence and 'bravery'...

I believe I've used all the points in the "note".

• This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post. - From Review Oct 17 '17 at 22:03
• Just to remind you, the purpose and the question is for the prince to deduce the color of his hat, and that's what I have described. I am suggesting an alternative theory to the blatant assumptions being made. Oct 18 '17 at 13:42
• It's a "fair test of intelligence and bravery": intelligence to find the test's solution, and bravery to be first to act on it. Nothing here precludes the intended answer from being a good fit. Your answer feels like Rules Lawyering to me—you have a desired path to a different solution entirely, and are gaming the question to make that path "fit" even though it is very evident what the intended solution is. Around here, that often results in: "This does not attempt to answer the posed puzzle, and so has been deleted. Puzzles with no lateral-thinking tag do not invite such answers."
– Rubio
Oct 19 '17 at 0:33
• In any case, you've elected to reopen a question that the community here recommended be closed for being "not an answer" (to this question). I was initially ambivalent but upon reflection feel I must agree, as you've chosen to ignore the "logical deduction" aspect entirely in your answer and thus sidestepped solving the actual posed puzzle. I invite you to reconsider, before the community elects to reaffirm their original opinion and adds a chorus of downvotes alongside.
– Rubio
Oct 19 '17 at 0:38
• @murphy1310, Rubio is trying to help you write better answers and not get lots of downvotes. Trying to smack him down for that is not going to win you any friends here. Oct 19 '17 at 20:07