Four Hats

Logic Level 3

Alice, Bob, Cathy, and David together have three white hats and four black hats. Without looking, they each put on one hat and throw the other three away. They then each look at their friends' hats (no one can see their own hat or the hats that were thrown away), with the following exceptions:

Alice cannot see Bob's hat.
Cathy cannot see David's hat.
David cannot see Alice's hat.

However, each friend believes the others can see three hats.

Alice says, "I know Bob doesn't know his hat color."

Cathy says, "I know David didn't know his hat color before Alice's statement."

David says, "Before Alice and Cathy's statements, I did not know that Alice did not know her hat color."

Given that the four friends are highly logical and made only true statements, exactly which people are wearing black hats?

Alice, Bob, and David Cathy and David Alice and David David Bob and Cathy Alice and Bob Bob and David Alice and Cathy

1 solution

Maggie Miller
Jul 17, 2015

Before any statements are made, the only way a person could know their own hat color is if they see three white hats (and therefore know their own hat is black).

Alice's statement tells us that she knows that Bob doesn't see three white hats. Therefore, Cathy and David cannot both have white hats.
Cathy's statement tells us that she knows that David doesn't see three white hats. Therefore, Alice and Bob cannot both have white hats.
David's statement tells us that he doesn't know whether or not Alice sees three white hats. Therefore, Cathy and Bob both have white hats.
Since Cathy has a white hat (conclusion 3), David has a black hat (conclusion 1).
Since Bob has a white hat (conclusion 3), Alice has a black hat (conclusion 2).

Thus, the answer is Alice and David .

I disagree with this answer; if we follow the original question's logic, the second and third statement fails, because it's always true except for bob, since each of them can only see 2 hats.

Details:

Alice's statement tells us that she knows that Bob doesn't see three white hats. Therefore, Cathy and David cannot both have white hats.

(I agree)

Cathy's statement tells us that she knows that David doesn't see three white hats. Therefore, Alice and Bob cannot both have white hats.

(Of course David doesn't sees three white hats, he can only see 2 hats in the first place, therefore it doesn't give an extra information.)

David's statement tells us that he doesn't know whether or not Alice sees three white hats. Therefore, Cathy and Bob both have white hats.

(Once again, since Alice can only see two hats in the first place, this statement doesn't give any extra information.)

"Given that the four friends are highly logical" - This suggests that each of them knows exactly who can see who's hat, so either add a statement saying that they don't know that each of them cannot see one other person's hat except bob or add an answer saying it's insufficient information. Or give an explanation as to what I'm missing.

A Former Brilliant Member - 5 years, 10 months ago

Oh yes, you're right - originally I let them each see all the other three hats but thought that would cause a problem. I'll re-edit; thanks

Maggie Miller - 5 years, 10 months ago

Four Hats

1 solution

0 pending reports