# Black and White Hats

JeffRosenspan
Posers and Puzzles 14 Dec '12 05:53
1. 14 Dec '12 05:53
Three men sit in chairs (#1, #2, and #3), in a straight line, all facing north, one behind another. Each man will have a hat placed on his head. No one can see his own hat at any time. The man in Chair #1 is in the front, he cannot see anyone else. The man in Chair #2 is behind Chair #1 and he can only see the man in Chair #1. The man in Chair #3 is behind Chair #2 and he can see both men in Chairs #1 and #2.

On the table, there are three blue hats and two white hats. Each of the three men will be randomly given one of those five hats and two will be discarded. No one will know the color of the discarded hats or the color of his own hat. The first person to use logic to determine what color hat is on his own head wins.

After five minutes, the man in Chair #1 stands up and says, "I win. I know the color of my hat."

The riddle is: What color was it, and how did he figure it out?

- Jeffrey Rosenspan
2. 14 Dec '12 12:002 edits
Blue.

If #1 and #2 were both wearing white, then #3 would have known immediately that he had blue himself. So, at least one blue hat is worn by either #1 or #2.

#2 deduces the same and still doesn't know his own colour, that means he is not seeing white in front of him that would give him the clue that his own hat is blue.

From the silence of #2, then #1 knows he is wearing no white hat.
3. 29 Dec '12 23:56
Sorry for being slow but where do the black caps come in?
4. Agerg
The 'edit'or
08 Jan '13 12:13
Originally posted by Habeascorp
Sorry for being slow but where do the black caps come in?
The blue ones are probably black on the inside.
5. 08 Jan '13 23:44
If I was 3 and could see two whites I might wait for 10 minutes or so for one of the others to make a fool of themselves, is this strategy flawed?
6. talzamir
Art, not a Toil
09 Jan '13 05:56
It reveals the hidden premises for sure. ^_^

Gamemaster: #1, you've had a minute to ponder. Any idea what you wear?
Contestant #1: no clue, I see nothing and no one has spoken.

Gamemaster: #2, you've had two minutes. What color?
Contestant #2: dunno

Gamemaster: #3, three minutes in, what do you wear?
Contestant #3: I don't have enough information.

Gamemaster: #2, four minutes in, what do you wear?
Contestant #2: it's either blue or white..

Gamemaster: #1, we are five minutes in. Do you know the color of your hat?
Contestant #1: yes.. logically I know, but since #3 is just waiting for me to make a total idiot of myself.. dunno.

Contestant #3: *snickers* gotcha
7. 09 Jan '13 23:25
Originally posted by talzamir
It reveals the hidden premises for sure. ^_^

Gamemaster: #1, you've had a minute to ponder. Any idea what you wear?
Contestant #1: no clue, I see nothing and no one has spoken.

Gamemaster: #2, you've had two minutes. What color?
Contestant #2: dunno

Gamemaster: #3, three minutes in, what do you wear?
Contestant #3: I don't have enough informatio ...[text shortened]... waiting for me to make a total idiot of myself.. dunno.

Contestant #3: *snickers* gotcha
I think it is flawed:

3 refuses to say anything even though he can see two whites.

2 Can see a white, since 3 says nothing, 2 says "my hat must be blue"

It is announced that 2 is wrong.

1 thinks, "For 2 to announce that he must have been sure, therefore he must think 3 is not answering because he can't see two whites, however if I was wearing blue then 2 would not have know what to say, therefore I must be wearing white.

1 quickly says white and 3 loses the game.
8. 10 Jan '13 22:23
Originally posted by iamatiger
I think it is flawed:

3 refuses to say anything even though he can see two whites.

2 Can see a white, since 3 says nothing, 2 says "my hat must be blue"

It is announced that 2 is wrong.

1 thinks, "For 2 to announce that he must have been sure, therefore he must think 3 is not answering because he can't see two whites, however if I was wearing ...[text shortened]... hat to say, therefore I must be wearing white.

1 quickly says white and 3 loses the game.
Of course as soon as 2 makes a statement I suppose 3 can say his hat colour, before 1 gets to hear whether 2 is right or not. That means 3 doesn't lose, so maybe the strategy is fine.

For perhaps an even more annoying strategy, 3 could pick his "delay" until saying the right answer from a random variable with something like the cauchy distibution, so his opponents have no idea how quickly he should answer and that may stump them completely, even if they have a clear 2 whites because they have no logical method to work out their hat colour.
9. 16 Jan '13 05:32
Originally posted by iamatiger
Of course as soon as 2 makes a statement I suppose 3 can say his hat colour, before 1 gets to hear whether 2 is right or not. That means 3 doesn't lose, so maybe the strategy is fine.

For perhaps an even more annoying strategy, 3 could pick his "delay" until saying the right answer from a random variable with something like the cauchy distibution, so h ...[text shortened]... they have a clear 2 whites because they have no logical method to work out their hat colour.
Its flawed because it is dependant on both #1 and #2 having white hats and the fact that the white doesn't know #1 or #2 will answer incorrectly as either one could be correct from a completely random guess.