3 Hats Riddle

Riddle courtesy of William Wu’s riddle archive.

There are 3 black hats and 2 white hats in a box. Three men (we will call them A, B, & C) each reach into the box and place one of the hats on his own head. They cannot see what color hat they have chosen.

The men are situated in a way that A can see the hats on B & C’s heads, B can only see the hat on C’s head and C cannot see any hats.

When A is asked if he knows the color of the hat he is wearing, he says no.
When B is asked if he knows the color of the hat he is wearing he says no.
When C is asked if he knows the color of the hat he is wearing he says yes and he is correct.

What color hat and how could he know?

Hint: Write down all permutations of hats and test them against the way each man answered.

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5 Responses to 3 Hats Riddle

  1. amy says:

    assuming they hear each other’s answer, it’s pretty simple. the only way a could know what color his hat is was if b and c were both wearing white hats. therefore, b and c are either both wearing black hats or one white and one black. b responded that he didn’t know either, that means that c had to have been wearing a black hat. if c had been wearing a white had, b would have known that he was wearing a black one, because they both could not have had white hates.

    :p

  2. Robert says:

    I also assumed that each man could hear the other answers and came up with the same answer as you.

    I’ve posted a harder riddle in this same vein. That one took me a while.

  3. miles says:

    but what if A was wearing a white hat, B was wearing a black and C was wearing a black. then A wouldn’t know because he saw one of each, B wouldn’t know because he see’s a black, and doesn’t know if he has a black or a white because both are possible… then also C wouldn’t know because no information was giving to him besides 2 i don’t knows.

  4. There are 7 possible combinations. We know there is at least 1 black hat being worn.

    A B C
    1 w w b
    2 w b w
    3 b w w
    4 w b b
    5 b w b
    6 b b w
    7 b b b

    If A sees two white hats, then he knows he is wearing a black hat since there are only 2 white hats. Since he answered “I don’t know”, this eliminates case 3.

    The possibilities are now:
    A B C
    1 w w b
    2 w b w
    4 w b b
    5 b w b
    6 b b w
    7 b b b

    If B thinks about this, he knows it means A does not see two white hats. A either sees one white and one black, or two black hats.

    If B looks at C and sees a white hat, he knows that he has a black hat. Since he also says “I don’t know”, this eliminates all the cases where C is wearing a white hat. This leaves us with:
    A B C
    1 w w b
    4 w b b
    5 b w b
    7 b b b

    If C thinks about it, he knows he is wearing a black hat.

  5. VR says:

    A B C
    -> facing this way

    b w w
    X w b
    X b w
    X b b

    a can see the above cases
    if B and C are wearing white hats A would have said yes. So option 1 is out.

    now B can look at C and he knows A’s answer.
    So he has to decide from the last three cases.
    if b saw a white hat in front he would have been sure that he is wearing a black one.

    But he said no. So C knows that he has to pick from cases 2 and 4. In both cases he wears a black hat.
    So the answer is BLACK.

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