# Hats and logicians

CalJust
Posers and Puzzles 01 Nov '14 15:41
1. CalJust
It is what it is
01 Nov '14 15:41
This is an old one, but still one of my favorites:

There is a vacancy for the job of Apprentice Logician and there are three applicants.

The Master Logician has the three candidates enter a room which has no mirrors. He tells them "I have here a box with three black hats and two white hats. In a minute I will blindfold you and put one of these hats on each one of your heads, removing the other two. Then I will remove the blindfold and leave the room. The first person to deduce (correctly) the colour of his own hat must knock on my door, tell me why, and he will get the job".

Since he is a very fair person and wants each to have an equal chance, he puts a black hat on each candidate.

The three look at each other,for a while and after about a minute one gets up and knocks on the door.

How did he reason?
2. wolfgang59
Mr. Wolf
01 Nov '14 23:46
Originally posted by CalJust
This is an old one, but still one of my favorites:

There is a vacancy for the job of Apprentice Logician and there are three applicants.

The Master Logician has the three candidates enter a room which has no mirrors. He tells them "I have here a box with three black hats and two white hats. In a minute I will blindfold you and put one of these hats on ...[text shortened]... r,for a while and after about a minute one gets up and knocks on the door.

How did he reason?
You give too much away.
The Master can put any hats on them and it is
possible for one Apprentice to deduce what hat he has.

The problem can be expanded to any number
ie N apprentices, N black hats, (N-1) white hats
3. CalJust
It is what it is
06 Nov '14 14:51
Originally posted by wolfgang59
You give too much away.
The Master can put any hats on them and it is
possible for one Apprentice to deduce what hat he has.

The problem can be expanded to any number
ie N apprentices, N black hats, (N-1) white hats
Agreed - so, any takers?
4. wolfgang59
Mr. Wolf
07 Nov '14 06:22
Originally posted by CalJust
Agreed - so, any takers?
I first heard this as gold and silver coins strapped to the forehead of
each. The first wise man says he does not know, the second wise man
says he does not know and the remaining wise man (who is blind!) gets it!
5. AThousandYoung
07 Nov '14 07:02
Originally posted by CalJust
Agreed - so, any takers?
6. CalJust
It is what it is
09 Nov '14 15:03
Here's a clue - a good logician (or philosopher, for that matter) would put himself in the other person's shoes, to find out how THEY would reason.
7. AThousandYoung
09 Nov '14 16:56
Next hint.

We see two black hats.

What are the possibilities?

1) I have a black hat.
2) I have a white hat.

What do the others see?

Either,

1) two black hats or
2) one white, one black
8. sonhouse
Fast and Curious
09 Nov '14 19:00
Originally posted by AThousandYoung
Next hint.

We see two black hats.

What are the possibilities?

1) I have a black hat.
2) I have a white hat.

What do the others see?

Either,

1) two black hats or
2) one white, one black
I gather the dude figured it out because the others could not, nor would he if he was in their shoes. Not sure what that leaves ME thoughđź™‚
9. AThousandYoung
09 Nov '14 22:55
Originally posted by sonhouse
I gather the dude figured it out because the others could not, nor would he if he was in their shoes. Not sure what that leaves ME thoughđź™‚
Any one of the three could have figured it out.
10. CalJust
It is what it is
11 Nov '14 15:07
Wolfgang and A1000Y seem to have figured it out, (or know this one).

11. Kewpie
since 1-Feb-07
16 Nov '14 07:12
I'm sure he's wearing a black hat, but I'm having trouble writing out the reasoning. If the other two don't know what they're wearing, they must be seeing just what he's seeing. But I don't think that makes sense.
12. CalJust
It is what it is
16 Nov '14 14:34
Originally posted by Kewpie
I'm sure he's wearing a black hat, but I'm having trouble writing out the reasoning. If the other two don't know what they're wearing, they must be seeing just what he's seeing. But I don't think that makes sense.
As I said, the Master is fair, so everybody had the same chance.

But that is not a good enough reason to say "my hat must be black".

The point of departure is to say: "I can only have a black or a white hat. Let's assume my hat is white. What would be the consequences?"
13. CalJust
It is what it is
20 Nov '14 09:13
Looks like nobody is interested, si I will give the answer anyway.

Let's call the applicants A, B and C.

A thinks: "I could have a white hat or a black one.

If I had a white one, B would see one white and one black hat.

So B would know that if he (B) had a white hat, then it would take C only a split second to figure out that he (C) had a black hat, since both white ones would have been accounted for. But C does not move, so B cannot have a white hat. So B would immediately know that he (B) had a black hat."

But B dies not move, hence A cannot have a white hat.

Ergo, A has a black hat.
14. talzamir
Art, not a Toil
22 Dec '14 23:52
Do things get problematic if an applicant realizes that it is not a co-operative game but a competitive one, and states

"Ah hah! I know what the color of my hat is", and head for the door if and only if he does NOT know what kind of a hat he wears?
15. 10 Jan '15 16:52
With two white hats and one black hat the tricksy black hatter can wait patiently for one of the white hatters to get up and announce incorrectly that his hat is black.

With one white hat and two black hats each black hatter cannot be sure that the other one is not tricksy, so neither declares.

With three black hats, none of the black hats can know that their hat is not white because if the other two stall it could also be one white and two blacks.