More Hat Logicians

Relevant wiki: K-level thinking

If Alice would have seen two black / two white hats, she would know, that her hat is of the other colour (white / black). This means that Alice had seen one black and one white hat.

Therefore, it is enough for Bob and Carol to look at each other's hat and they can be sure, that their hats is of the other colour.

Hence, our answer should be:

$\boxed { \text { Yes, they can. } }$

If Alice says, "I do not know the color of my hat," it means she sees a white hat and a black hat. Therefore, either Bob or Carol can see two hats of the same color and determine what color their hat is. Then, the other one can say their hat is the other color. Therefore, the answer is "Yes, they can."

$\text{UNCERTAINITY for AlLICE:}$

$\text{at first, if Alice would have seen two black hats,then he would have been sure that he was wearing a white hat on his head.}$

$\text{as there were only 2 black and 2 white hats.so, same for the white hats.}$

$\text{that means, Alice saw different colors of hats on their heads.that's why he was in doubt/not sure}$

$\text{ACKNOWLEDGEMENT for CAROL \& BOB:}$

$\text{Bob's view:as Bob know that his and Carol's hats color is different.and as there are only 2 colors.so, among-st 3 ,two of them wore}$

$\text{the same colors hat.so, if Alice's and Carols hats color is same,then he wore the other one.if his and Carol's hats color is different}$

$\text{then,he wore same colors hat as Alice.}$

$\text{same view for Carol}$

$\text{that's why --yes,they can identify their hats color.}$

Saying she doesn't know tells the other two that she sees a black hat and a white hat, so Bob and Carol know they are wearing the opposite colour of hat to the other.

Interestingly, by saying she doesn't know what colour her hat is, Alice can never deduce her hat colour, but if she had kept quiet, she could have.

One of Bob or Carol can see two hats of the same colour, and would have been able to say their hat colour without anyone else saying anything, letting Alice and the other person know that they had identical hats, so all three would be able to deduce their hat colour. By speaking, Alice can't know how Bob and Carol know their hat colour, so cannot guess her own.

There are already other answers that explain this problem perfectly well, but I think it is interesting to note the logic used in them is not required.

There are 3 people and only 2 of each color hat. That means that the group of 3 has 2 people with the same color hat, and 1 person with a different color hat. The person with the different color hat will see the two other people having the same color, and will know what color they have by simple elimination. Alice's statement is unnecessary.

If Alice does not know the color of her own hat, that implies that both Bob and Carol have different color hats. If they had the same color hats, Alice would know that she must have the opposite color. This, since they now know that they have different color hats, whatever Bob has, Carol knows she has the opposite, and vice versa.

Bob and carol can ask each other what color hat they're wearing.

Since they are always logical and always tell the truth, they will respond with the correct answer.

Alice didn't knew what was her hat color ,means other two are having their hats with different colors, this shows that any two of the three are having hats of same color. this will allow one of them(BOB or CAROL) to identify their hat color by seeing the same color of other two. Let BOB identified his hat color( means ALICE & CAROL are having same color). Now seeing BOB identifying his hat color CAROL will become sure that BOB was able to identify his color only because he was able to see two same colored hats.... now she(CAROL) can identify her color by seeing the color of BOB.....

সহজ! এলিস এর কথা অনুযায়ী বব আর ক্যারোল বুঝতে পারে যে তাদের দুইজনের মাথায় ভিন্ন রঙ এর হ্যাট আছে। (কারণ একই রঙ এর হ্যাট থাকলে(মনে করি দুইজনের মাথাতেই সাদা রঙ এর হ্যাট ছিলো) এলিস বুঝে যেতো তার মাথায় বাকি থাকা অন্য রঙ এর হ্যাট আছে(মানে কালো রঙ এর হ্যাট আছে)

যাই হোক,যেহেতু বব আর ক্যারোল জানে তাদের মাথায় ভিন্ন রঙ এর হ্যাট আছে,বব ক্যারোল এর মাথায় হ্যাটের রঙ দেখে বুঝতে পারে তার মাথায় অন্য রঙ এর হ্যাট আছে(মানে যদি বব ক্যারোল এর মাথায় কালো রঙ এর হ্যাট দেখে থাকতো,তাহলে সে বুঝতে পারতো তার মাথায় সাদা রঙ এর হ্যাট আছে)

সুতরাং,সহজ উত্তর! তারা সহজেই বুঝতে পারবে তাদের মাথায় কি রঙ এর হ্যাট আছে!

Alice must of seen a black and white hat or she'd know the colour of hers as if there were two of the same colour, only one colour would be left. So through seeing the other hat, bob and Carol could see what colour theirs is black or white.

Now, bear with me. I may be simple minded in saying this, but couldn't they just ask each other what colour their hat is? For example, Bob could ask Carol, "What color is my hat?" Carol could then respond with, "The color of your hat is __ ." Carol and Alice could then proceed to ask the same question. Wouldn't that be a much simpler solution? No where are there any rules in which it says they may not convey their known information.

It says they can only see the others hats, not their own.

So, Alice opened the box and saw 3 hats. 2 of one color, 1 of another color. - I dont know which color my hat is, she said. Bob and Carol doesnt know their color either, but from what alice told them she saw, they can compare what they are seeing themselves. With all three of them seeing different things, discussing and comparing - they can figure out that there are in fact a total of 2 black hats, and 2 white hats. Now when looking in again you will know what color hat it is you are not seeing. That's your hat buddy.

The problem should add a note that "no one should tell others their hat colors." Because I'm a realist and I'd tell them to look at each other's hat and tell them what their hat color is.

A sees one Black and one White. She has only 50% chance of getting it right so she says no to be safe (logicians do not play Lottery!). A less logical person would had taken a chance. Heck! It is just a game

As Alice is confused means that Bob and Carol have opposite coloured hats, means Bob sees Carol's hat and knows his hat color and same with Carol