Arun, Bob and Charlie know that there are 3 red hats and 2 blue hats in a dark room. Each enters the dark room, picks up the first hat he gets his hands on and wears it.

After they come out of the dark room, they can't see their own hats. They are curious whether they can figure out the colour of their own hats only by looking at each other's.

Arun looks at Bob's and Charlie's hats and says that he can't determine the colour of his hat. Bob looks at Arun's and Charlie's hats and says that he is clueless as well. After Bob finishes saying that, without even looking at Arun's and Bob's hats, Charlie exclaims, "I know the colour of my hat!"

Which colour hat is Charlie wearing?

[SOLVED]

5 comments

Shreemoyee Sarkar solved this puzzle:

Possible cases (Order: Arun's hat, Bob's hat and Charlie's hat) are:

  • B B R
  • B R B
  • B R R
  • R R R
  • R B B
  • R R B
  • R B R

R B B is eliminated because there are only two blue hats but Arun can't decide which hat he has.

B R B and R R B are eliminated because Arun cannot decide and so Bob knows that both Bob and Charlie do not have blue hats. So, if Bob sees a blue hat on charlie, he can say that he has a red hat.

The only remaining cases are:

  • B B R
  • B R R
  • R R R
  • R B R

Now looking at the cases that passed, we see that only red hat is possible for Charlie.

Indhu Bharathi solved this puzzle:

Charlie is wearing a red hat.

Of the possible combinations:

  • B B R: Stays.
  • B R B: Ruled out. Otherwise Bob would have found the answer since there are only two blue hats.
  • B R R: Stays.
  • R B B: Ruled out. Otherwise Arun would have found the answer since there are only two blue hats.
  • R B R: Stays.
  • R R B: Ruled out. When Bob tries to guess his hat, he sees R X B (where X is unknown). If X was Blue, Arun can guess his hat since there are only two blue hats and Arun's hat must be red. The fact that Arun isn't able to find the answer means Bob's hat is not blue. Bob knows this and so he can answere. But according to the question, B isn't able to find out the color of his hat. It implies that this is not the combination they were wearing.
  • R R R: Stays.

In all the options that stay, Charlie is wearing a red hat.

Don Del Grande solved this puzzle:

If Bob and Charlie both had blue hats, Arun would know his hat is red; therefore, at least one of Bob and Charlie's hats is red.

Bob, having heard Arun, knows that Bob and/or Charlie have a red hat. If Charlie is wearing a blue hat, then Bob would know that his own hat is red.

Since he doesn't, Charlie's hat must be red.

Mike Fee solved this puzzle:

Assumption: Arun, Bob, and Charlie are all good at logical thought.

If Arun could see two blue hats he would know that his was red, as it would be the only remaining choice. This doesn't happen. So, Bob knows immediately that he and Charlie either have (red, red) or (red, blue) or (blue, red) hats respectively. If Bob could see a blue hat on Charlie's head then Bob would immediately know that his hat must be red. This doesn't happen, so Charlie's hat must be red.

Kunal Kishor solved this puzzle:

Seems that Charlie just became a communist!

The clue is that it is impossible to determine the color of one's hat only when he sees at least 1 red on another head (if both are blue, he's wearing the RED).

Now,

  • Andrew couldn't determine his own color. It implies that at least one of Bhushan or Cat had a RedHat installed.
  • Bob couldn't determine his color (and he's not dumb!): If Cutie had a BLUE hat on, then the "at least one redhat" which Arun saw would be his own! And he would have exclaimed immediately!

But he didn't which means that our "C" just went RED making it impossible for B to determine his own color.

Well C didn't even had to look at others' hats!

Credit

This puzzle is taken from folklore.

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