Hats

[Desardes]

Here's another question but keep in mind of the details that i include.
There are three prisoners awaiting execution when suddenly the prison ward comes in and stops the execution. (A little backgroud of the three prisoners: they know each other extremely well and we are too assume they're incredibly intelligent). The prison ward offers them a way of avoiding executing and they agree to it.
They all sit in chairs in a vertical line with the second only seeing the back of the first and with the third seeing the back of the second. The prison ward informs them that he has 2 black hats and 3 whitehats and he will place a hat on each prisoner, hence, the scenario being that the 3rd knows that the second is wearing and the second knowing what the third is wearing and the first UNAWARE of anything. They're given a minute and whoever calls out the colour of their own hat will determine the fate of the three altogether. (So, if someone screams out something and he's wrong, they all die but if he says it right, they all go free). After 59 seconds, the First one screams the correct colour of his hat and they go free. How did he know and what was the colour of the hat?

[Desardes]

Remember, after 59 seconds the FIRST onme answered. So what would that imply? Secondly, they all know each other to be geniuses...........

[joeschmoe]

ok I'm replying, so if you don't want to know the answer I got, then don't read below.
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I'm assuming that you mean that the 3rd guy can see BOTH hat 1 and hat 2, and that the 2nd guy can only see hat 1, and that the 1st guy can see nothing. you didn't explicitly say that the 3rd guy could see the 1st guy too, just that he could see the back of the 2nd guy, but unless the 3rd guy can see both, I don't know how to do it.

the guys decide to give each other ~20 seconds to think about the situation, starting with the guy in the back. this means that the guy in the back has seconds 0-20 to say the color of his hat if he can deduce it. If he can't (and thus doesn't say anything), the second guy has seconds 21-40 to try to deduce his hat color. If he says nothing, the first guy has seconds 41-60 to deduce his hat color.

starting at the back, the 3rd guy can see 4 possible situations. I'll represent these with comma separated lists, where the first letter means the color of the first guy's hat, the second letter means the color of the 2nd guy's, and the last letter represents the last guy's hat color (or a question mark if he doesn't know).

4 possibilities, seconds 0-20:
1) w,w,? (guy 1 has a white hat, guy 2 has a white hat, and guy 3 doesn't know)
2) w,b,?
3) b,w,?
4) b,b,w (the 3rd guy knows his hat is white because there are only 2 black hats and he sees them both)

since the 3rd guy in the riddle didn't speak up during his 20 second time limit, he must not have seen 2 black hats.

so our scenario must be one of the first 3 possibilities.

now, on to guy 2.

2 possibilities, seconds 21-40:
1) w,? (there were 2 possibilities of the 3 above where guy 1 had a white hat, so guy 2 can't know his hat color in this case)
2) b,w (the 2nd guy knows his hat is white, because the only valid possibility of the 3 where guy 1 has a black hat is option 3 from above)

since the 2nd guy in the riddle didn't speak up during his 20 second time limit, he must not have seen 1 black hat.

now, on to guy 1.

if no one has spoken up by the 40th second, guy 1 must have a white hat. I supposed he waited till the last second to make sure one of the other geniuses hadn't made a mistake.

[unbiased]

Hi,

Can the third person see the hat of the first person?

Originally posted by Desardes
Here's another question but keep in mind of the details that i

[Desardes]

thanks for mentioning that unbiased. and yes, he can

[unbiased]

I think I also have the answer.....don't read if you don't want to see it.













Ok, these guys are smart, so they should know quickly the answer if the answer were obvious.

b
b

would be immediately obvious to the #3 guy, so this can't be the correct answer because he would have said it immediately.

so we are left with
b w w
w b w
? ? ?

the ?'s are what the #3 guy is wearing and they really don't matter...nobody would ever see those anyway

-since #3 did not know the answer right away, everybody would be smart enough to figure out that either #1 or #2 or both are NOT wearing a black hat.
Thus if #2 saw a black hat, he would know immediately that he were wearing a white hat...but that situation did not happen...and this is a smart guy...so he must still be in doubt of what he is wearing.

so we are left with
w w
b w

-So since #3 and #2 did not immediately know the answer, and they are all smart guys, then the only possibility would be for the #1 guy to know that he was wearing a white hat - which is the only possibility that would not be immediately obvious to either #3 or #2.

I'm pretty sure that is correct.
Is it?

Thanks,
Unbiased


Originally posted by Desardes