# Variation on the Hats Puzzle

CalJust
Posers and Puzzles 10 Oct '06 14:41
1. CalJust
It is what it is
10 Oct '06 14:41
Here's another hat-wearing puzzle which some of you may know. If you do, give the others some time to work it out.

Three guys apply for the job of Logician with the Wise Mens Club.

The Chief guru comes in and tells them the following:

"Here in my box I have five hats: three are black and two are white. I will now blindfold you and place a hat on each of your heads and put the other two back in the box. When I remove the blindfolds, each of you will see the other two hats, but not your own. Whoever deduces the colour of his own hat, can come into my office and explain how you figured it out. If you are right, the job is yours."

It should go without saying that there is no mirror in the room.

With that he blindfolded them and placed a black hat on each head. (What could be fairer than that?!)

When the blindfolds were removed, after about three minutes person A got up and explained to the Boss his thinking.

How did he reason??

(Sorry about the sexist imagery. But then, generally speaking, men are more logical than women ;-) )

I prefer this puzzle to the Four Hats one because here each one has an equal chance, and, this time round, definitely the best man wins. In the former, the middle guy had a clear advantage.

In peace

CJ
2. PBE6
Bananarama
10 Oct '06 15:18
Originally posted by CalJust
Here's another hat-wearing puzzle which some of you may know. If you do, give the others some time to work it out.

Three guys apply for the job of Logician with the Wise Mens Club.

The Chief guru comes in and tells them the following:

"Here in my box I have five hats: three are black and two are white. I will now blindfold you and place a hat on eac ...[text shortened]... ely the best man wins. In the former, the middle guy had a clear advantage.

In peace

CJ
I think this has been posted before...but here's the solution anyway:

Person A sees two black hats. Now he reasons that if he were wearing a white hat, either of the other two would reason thusly: "I see A wearing a white hat, and B (for the sake of simplicity) wearing a black hat...if I too (person C) were wearing a white hat, then B would immediately know he was wearing a black hat and report it...however, he has not done so, so I must have a black hat on and I shall report it." However, C did not report the colour of his hat either, to which A reasoned that he himself could not be wearing a white hat, so his hat must be black.

However, I'm not sure if there isn't some bootstrapping going on here. The other two men's silence is the trigger that leads person A to report his hat as black, but how can he be sure that the silence means no one can identify their hat with the information given and not that the other two are simply slow in their reasoning?
3. uzless
The So Fist
10 Oct '06 17:50
Originally posted by PBE6

However, I'm not sure if there isn't some bootstrapping going on here. The other two men's silence is the trigger that leads person A to report his hat as black, but how can he be sure that the silence means no one can identify their hat with the information given and not that the other two are simply slow in their reasoning?
Indeed, the silence should indicate that there is not enough information for person B and C to decipher what hat they are wearing. We are to assume these other two "Logicians" are equally adept intellectually as our person A.

Therefore, it should be a tipoff to person A that no one can figure it out.
4. PBE6
Bananarama
10 Oct '06 18:17
Originally posted by uzless
Indeed, the silence should indicate that there is not enough information for person B and C to decipher what hat they are wearing. We are to assume these other two "Logicians" are equally adept intellectually as our person A.

Therefore, it should be a tipoff to person A that no one can figure it out.
But if they're equally adept, all three men would be able to answer at the same time since the problem is symmetrical. I still think it's a bit squirrelly...
5. uzless
The So Fist
10 Oct '06 18:48
Originally posted by PBE6
But if they're equally adept, all three men would be able to answer at the same time since the problem is symmetrical. I still think it's a bit squirrelly...
Yes, it assumes that one person is smarter than the other 2.

Should person A had a white hat, it may have been that the other two were just slow at figuring out the answer and that's why they didn't go tell the headmaster the answer.

It's hypocritical to assume the other two are dumber if he's wearing a black hat, yet not assume they are dumber if he is wearing a white hat.
6. CalJust
It is what it is
11 Oct '06 10:45
Originally posted by uzless
Yes, it assumes that one person is smarter than the other 2.

Should person A had a white hat, it may have been that the other two were just slow at figuring out the answer and that's why they didn't go tell the headmaster the answer.

It's hypocritical to assume the other two are dumber if he's wearing a black hat, yet not assume they are dumber if he is wearing a white hat.
Both PBE6 and uzless miss one important point: it does not take a lot of logic to figure out that anyone who sees two white hats, can in a microsecond deduce that he must have a black hat. No thinking required!

Because this does NOT happen, it is clear that nobody can see one white and one black. The first one to figure this out must be smarter than the other two.

As far as symetricality is concerned, why doesn't everybody on RHP figure out a any puzzle at the same time?? All have an equal chance to do so!!
7. PBE6
Bananarama
11 Oct '06 17:40
Originally posted by CalJust
Both PBE6 and uzless miss one important point: it does not take a lot of logic to figure out that anyone who sees two white hats, can in a microsecond deduce that he must have a black hat. No thinking required!

Because this does NOT happen, it is clear that nobody can see one white and one black. The first one to figure this out must be smarter tha ...[text shortened]... everybody on RHP figure out a any puzzle at the same time?? All have an equal chance to do so!!
I think it's fairly clear that we both understand the problem and the solution. However, your statements do not address the main problem I have with this question, namely: The other two men's silence is the trigger that leads person A to report his hat as black, but how can he be sure that the silence means no one can identify their hat with the information given and not that the other two are simply slow in their reasoning?

I agree that if one person saw two white hats, then he would immediately report his hat as black. I do not agree that if a person saw one white hat and one black hat that they could quickly surmize that they were wearing a black hat due to the others' silence. This step involves visualizing a hypothetical situation and 2 other people's presumed responses to that situation, and I think you'd have to agree this would take some amount of time (it's your "brain teaser", after all...I don't think you'd post a trivial puzzle for fun). With that implication in mind, here are 2 possibilities:

1. Persons A, B and C are all equally adept at creative and logical thinking, and would be able to work out the problem in the same amount of time.

2. Person A is better at creative and logical thinking than either B or C, and can therefore figure out the problem before them.

In case 1, all persons would come to the same conclusion simultaneously. In case 2, person A could not be sure where B and C's were stumped. Seeing either one black hat and one white hat or two black hats would still require B and C to think, and that takes time. Unless A had a certain measure of the amount of time B or C require to crank out the solution, he could not be sure of his own hat colour.

In answer to your final statement, puzzles posted on RHP are not generally symmetrical. But don't tell Phlabibit, you'd crush him (he's still waiting to solve one ðŸ˜‰).
8. Palynka
Upward Spiral
13 Oct '06 15:341 edit
Originally posted by PBE6
I think it's fairly clear that we both understand the problem and the solution. However, your statements do not address the main problem I have with this question, namely: [b]The other two men's silence is the trigger that leads person A to report his hat as black, but how can he be sure that the silence means no one can identify their hat with the information . But don't tell Phlabibit, you'd crush him (he's still waiting to solve one ðŸ˜‰).[/b]
Thing is, the fact that, even if A was wrong, he was able to deduce all that faster than any of the other two could deduce even the first step. A's logical reasoning would have failed only because the other two failed to reason logically.

Therefore, A would get the job anyway. ðŸ˜‰
9. sven1000
Astrophysicist
13 Oct '06 16:56
Perhaps they were all equally bright and logical, but person A was stronger as well, so muscled the other two out of the way as they all went for the door!
10. PBE6
Bananarama
13 Oct '06 16:57
Originally posted by Palynka
Thing is, the fact that, even if A was wrong, he was able to deduce all that faster than any of the other two could deduce even the first step. A's logical reasoning would have failed only because the other two failed to reason logically.

Therefore, A would get the job anyway. ðŸ˜‰
True. And life's a gamble anyway, right? But do you think I'm spinning myself in cirlces, or is the solution not completely coherent?
11. Agerg
The 'edit'or
13 Oct '06 17:124 edits
Originally posted by PBE6
True. And life's a gamble anyway, right? But do you think I'm spinning myself in cirlces, or is the solution not completely coherent?
see... if any of them sees a white hat, then that person would know that if his hat was also white then somebody else wins by virtue of their reflexes, perhaps that was the same conclusion reached by all participents at the same time...that there is at least a moment of silence means that there is no such quick response, and so it is safe to say that none of them can see a white hat...it was stated in the OP that the best man wins....A could just as well have been B or C (best could be described as simply as *being in the best position such that their answer was received first*)
12. Palynka
Upward Spiral
14 Oct '06 08:28
Originally posted by PBE6
True. And life's a gamble anyway, right? But do you think I'm spinning myself in cirlces, or is the solution not completely coherent?
In a way, yes, it certainly isn't bullet-proof as it requires an assumption even if such an assumption sounds credible to me.