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Sunderhorn
09-24-2008, 11:55 AM
This was posted in my realm forums:
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Somewhere in the universe there exists a world in which people either have green or purple nostrils; in all other respects, though, they are like us. There are no mirrors, no reflective surfaces, and no photographs. This means that they can never see their own nostrils. In fact, if they are ever to discover the colour of their nostrils, they will, without fail, commit suicide that evening at midnight.

I should note that everyone within this society is both completely honest (meaning anything anyone says is the truth) and perfectly capable of logic (and thusly logical reasoning).

Now riddle me this:

There exists a home in which there are three roommates, all three of whom have green nostrils. (Though, of course, each only knows that the other two have green nostrils.)

One day, they order a pizza. Now, the person delivering the pizza is not having a good day, and after receiving a low tip from the three roommates, this person looks at the three, and says:

"At least one of you has green nostrils."

He then turns around, and walks away.

How many days pass until the 3 roommates all commit suicide?

(You have to be able to EXPLAIN how you came up with your answer. Don't just blurt out numbers.)
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Anybody have any thoughts? The post said the answer had to do with an ethical dilemma between the roommates, and having wracked my brain over this I came to the conclusion that none of them can deduce the color of their own nose.

Figured I'd post here to see what some of your super-geniuses thought.

My best answer for any kind of ethical dilemma is that given the context, all three roommates know there is only one likely reason why the pizza guy made the comment.

Turkson
09-24-2008, 12:06 PM
commit suicide that evening at midnight.

perfectly capable of logic (and thusly logical reasoning).



Well, there's your problem right there.


Since this is a theory game, I'm going to go with they all stay alive. The pissed off delivery boy said "At least one of you..." The three will look at each other and note his fellows have the requisite colored nostrils. Since these are logical, rational being (and roommates) they're not going to mention anything about nostril color as it would lead to the suicide of a fellow. Each of the three will go on satisfied in the knowledge that the delivery boy was right and no one died. Alot of assumptions on my part, and I kinda surprised myself with the optimism our green nostril'ed test subjects would show.

Do I get a cookie?

Gwyndolin
09-24-2008, 12:31 PM
Each individual solves the problem seperatatly, from their perspective they are Person C.

Person A is known to be green.
Person B is known to be green.
Person C assumes he is Purple.

Person A sees 1 Green and 1 Purple.
At this point the comment is fulfilled and he cannot determine his own color.
At Midnight Person B doesn't kill himself. (He also saw could not determine his color.)
Day 1 has passed - No deaths.
Because Person B didn't kill himself, Person A knows he can't be purple, else B would have known his own color the first night and killed himself. Likewise B knows he can't be purple.
Day 2 has passed - A and B kill themselves if C is purple. No one kills themselves yet.
Because A and B didn't kill themselves on Day 2, C knows he cannot be purple.
Day 3 All three know they are green and kill themselves.

Sunderhorn
09-24-2008, 12:37 PM
Funnily enough, as Knoway from Perenolde pointed out, it doesn't matter what color person C's nose is....he winds up dead anyway [just a day later than the other 2].

Horacio
09-24-2008, 12:44 PM
They are at an age where there are, presumably, adults living in an apartment our house together. There is not mention in the original question posed whether this condition is congenital, sudden onset brought about by exterior stimuli, etc. I find it hard to believe they've lived all thier lives with no one ever revealing to them the color of thier nostrils, requiring a vindictive pizza boy to enlighten them and pose such a question.

How does the pizza boy know his mention of the color of thier noses will cause grief and potential suicide if he is not himself aware of his own nose color?

This is just....bizarre. There's too many holes, too much information left out....too much social interaction that mus take place between people without folks dropping dead left and right all over the place.

Satrina
09-24-2008, 01:08 PM
The logic is flawed since it allows boundary cases:

One of Person A/B/C, or two of them, or all three at the same time exclaim "well, thank $deity it isn't me!", to which the other persons, being completely honest, say "well, um...". Death occurs at most 1 day later, depending on what time the pizza boy came.

The completely honest clause also pretty much guarantees that the race would have self-exterminated not long after language was invented anyway.

Alent
09-24-2008, 01:13 PM
Somewhere in the universe there exists a world in which people either have green or purple nostrils; in all other respects, though, they are like us. There are no mirrors, no reflective surfaces, and no photographs.

In such a case... how does the pizza guy get there? W/o reflective surfaces, everything is pitch black.

Sunderhorn
09-24-2008, 01:18 PM
link to headache

The blue-eyed islanders puzzle What’s new (http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/)

Tobius
09-24-2008, 01:27 PM
I think none of them die as a direct result of the pizza boy incident.

They would all know that by logically thinking through the possible implications of the pizza boy's statement that there would be two potential outcomes:
They find out their nasal colouration and kill themselves
They don't find out their nasal colouration and don't kill themselves


By choosing (or logically reasoning) not to think the problem through all the way as Gwyndolin does they don't know their colours and so survive - resulting in the best case scenario. Thats logic :)

Edit: for clarity

Sunderhorn
09-24-2008, 03:04 PM
Not thinking has always been the easiest way to deal with any problem, that is true. ;)
It's also usually the most expensive way.

I actually enjoyed this little teaser, haven't werked my brain this hard since I tried opening an aspirin bottle last week.

Alent
09-24-2008, 03:05 PM
I maintain the pizza boy could've never seen the color of their nose to begin with, because of the lack of reflective surfaces.

Sunderhorn
09-24-2008, 03:12 PM
Maybe the pizza boy was really a marmot wearing a pizza boy suit? I know this one who sees me every time I visit this site. I have nightmares about his cold stare...

Tobius
09-24-2008, 03:40 PM
I maintain the pizza boy could've never seen the color of their nose to begin with, because of the lack of reflective surfaces.

Good point! They also wouldnt be able to see each others noses either ;)

Leeroi
09-25-2008, 11:38 AM
Gwyndolin's work assumes that each person "knows" that the other person worked out their own nostril color, which is wrong. Knowing that at least one person has green nostrils, each person will say to themselves, "If any of us roommates see two people with purple nostrils, they will know they must be the "at least one" green and kill themselves. I see two people with green nostrils, so I could still have purple nostrils. Since both of the other roommates have green nostrils, each of them will see one person (each other) with green nostrils (since mine are purple) and they will not kill themselves." The malicious comment from the marmot-cum-pizzaboy is fulfilled by everyone without anyone getting any knowledge concerning their own nostrils (i.e no one dies and that doesn't cause a problem for anyone other than the sick-bastard marmot). The thin-skinned, greedy marmot has not provided any new information to the roommates. Thus nothing changes and all three live.

What is interesting, is if the pizza-loving-marmot had said "at least 2 of you have green nostrils". Then, they would all die on the second midnight. When they woke up the morning following their fateful pizza bash and saw their other two roommates alive, they would know that their own nostrils must be green since the two other roommates did NOT kill themselves. I.e. each roommate will say to themselves: "If I had purple nostrils, the other two would know it was them and they would both have killed themselves last night; since they did not, they must each see 2 roommates with green nostrils, and so I must have green nostrils also." Since all three would come to this realization after the first midnight, they would all kill themselves on the second midnight -- but only after having a little "chat" with the pizza delivery marmot.

But, the marmot wasn't aware of this, gave a statement with not enough information to change the living arrangement for the roommates, deserves his low tip, and had better look out the next time those boys order pizza....

Alent
09-25-2008, 03:58 PM
A friend of mine points out that for the pizza delivery boy to say such a thing, our green nostril'd friends must be REALLY lousy tippers.

Sunderhorn
09-25-2008, 04:51 PM
Leeroi, the pizza boy does provide new information in the form of common knowledge, and they kill themselves 3 days later. This puzzle is ridiculously hard to think through for normal people like me, but that's the correct answer.

Kazeyonoma
09-25-2008, 05:08 PM
I don't understand how that can be true.

This is how I thought it out with my co workers.

Possibilities:
3 Purple 0 Green
2 Purple 1 Green
1 Purple 2 Green
0 Purple 3 Green

Night one:
The 3 aliens look at each other and see that at least sees 1 green. Before they go to sleep, each person can already conclude that the top 2 possibilities of 3 purple or 2 purple are impossible (Although they don't know that the others come to the same conclusion). No one dies, they go to sleep and wake up

Night two:
The 3 aliens think, why didn't anyone die. Alien A says, well Alien B must've seen Alien C and thus didn't commit suicide, likewise Alien C saw B and didn't commit suicide, Alien A's color is inconsequential at this point. LIKEWISE, the Alien's B and C do the same thought process and again, are inconsequential to their own world. What new information is gained here? None. All 3 go to sleep still with the possibility of 1 Purple 2 Green or 3 Green (in each one's minds)

Night three:
Having gained no new information, from the previous night, they are in the same boat as the night before. No one dies. And no new information is gained.

So where does it occur that they gain some knowledge after the first night and thus reach a suicidal conclusion? They never can move past the fact that there are 2 possibilities always (for their own conclusions): 1 purple and 2 green, or all 3 green (the variable person is of course themselves)

Elaborate please Sunder?

Sunderhorn
09-25-2008, 05:37 PM
Kaz, on the second night, they know that if there are only 2 of them with green nostrils, those two will kill themselves on that night. Any given roommate will look at the other 2 and say "if they each see only 1 other person with green nostrils, they will have to kill themselves tonight, since they will know that only 2 of them has green nostrils".

The knowledge they gain from the pizza marmot is common knowledge (http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29)

A and B here refer to any of roommates [this was copied from a comment on another site]:

"The new information (for A) is not that somebody has green nostrils. A already knows that. The new information is that B (whom A already knows to have green nostrils) also knows that somebody else has green nostrils."

Leeroi
09-26-2008, 08:44 AM
That still makes no sense. The following logic applies to any given roommate.

To stay with the assignments given by Gwydolin, C is any given roommate and A and B are the "other" two roommates:

(1) C knows that the criteria of "at least one" is met regardless of what his own color is, since A and B both have green nostrils. Thus C does not know what his color is and will not kill himself.
(2) C knows that the "at least one" criteria is met for A since A will see that B has green nostrils, and the criteria is met for A regardless of C's color. Thus C knows A will not kill himself and C knows nothing of his own color.
(3) C knows that the "at least one" criteria is met for B since B will see that A has green nostrils, and the criteria is met for B regardless of C's color. Thus C knows that B will not kill himself and C knows nothing of his own color.

So: C does not die and C knows that neither A nor B will die. C also knows that his own color is irrelevant to the fate of any of the roommates - all will continue to live because C can see that both A and B have green nostrils.

Since the "C" perspective is true for all roommates, all roommates will (1) not kill themselves and (2) know that their roommates will not kill themselves either, and (3) that his own color makes no difference in this outcome. So... no one dies. Not the first night, not the second night, not any night, until the evil marmot (is that repetitive?!?) comes back and clarifies that "at least 2 of the roommates have green nostrils".

*** Edit ***
I just noticed that a roommate will not kill themselves if they konw they have green nostrils, but rather "if they are ever to discover the colour of their nostrils". So knowing that you have purple nostrils is just as lethal as knowing you have green. I *believe* the above logic still holds and that C will continue be blissfully ambiguous as to his own nostril colors, but I need to think about that for a bit....

Sunderhorn
09-26-2008, 09:07 AM
The link in my previous post is probably the best explanation you're going to get.

Here's 2 more links:
puzzle (http://www.xkcd.com/blue_eyes.html)
solution (http://xkcd.com/solution.html)

But honestly, the wiki one was the most helpful to me once I read through it carefully like 5 times :p.

Kazeyonoma
09-26-2008, 09:11 AM
I still don't see it despite reading your post several times, I am still seeing it from Leeroi's PoV.

Kazeyonoma
09-26-2008, 09:14 AM
Kaz, on the second night, they know that if there are only 2 of them with green nostrils, those two will kill themselves on that night. Any given roommate will look at the other 2 and say "if they each see only 1 other person with green nostrils, they will have to kill themselves tonight, since they will know that only 2 of them has green nostrils".

The knowledge they gain from the pizza marmot is common knowledge (http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29)

A and B here refer to any of roommates [this was copied from a comment on another site]:

"The new information (for A) is not that somebody has green nostrils. A already knows that. The new information is that B (whom A already knows to have green nostrils) also knows that somebody else has green nostrils."

I don't think your first statement is true.
why does "if there are only 2 greens" ever apply? There are 2 scenarioes, 2 greens or 3 greens right? but they don't know for sure. C can assume that A is trying to deduce this as well, and if C assumes that A sees 2 greens, then yes you are green and you die, BUT that is assumption, likewise C can assume that A sees 1 green and 1 purple, and he still wouldn't commit suicide right? So you can never complete that statement.
I think the assumption is being made that C can assume that A sees 2 green and thus must say "at least 2 green" but since A cannot assure C that that is what he sees, C can never come to that conclusion.

Satrina
09-26-2008, 09:27 AM
This problem is quite different from the blue islanders one. Edit: Except that it's not!

Logic solution: Edit: Yeah, I missed a piece of logic and the solution below (and Gwydolin's on page 1) is right.

Social solution: As I gave before. This one applies because of the redundant honesty clause that has no other impact on the problem other than to open the door to that solution. Edit: This one still holds :P

Kazeyonoma
09-26-2008, 09:32 AM
Thank you Satrina, my brain can rest at ease.

Sunderhorn
09-26-2008, 09:51 AM
It's a tricky one to be sure. I'll try to channel the All-Knowing Marmot and see if I can try to type a clear answer. Remember all the roommates are logical. As Gwydolin pointed out, each roommate goes through this same thought process:

1 - The first step is to realize that if only one roommate had green nostrils, he would kill himself on the first night. They all know this.

2 - Obviously, everyone is still alive on the second day. Each one looks at the 2 roommates and realizes that if they are the only 2 with green nostrils, those two will kill themselves that night. Because if they are the only two with green nostrils, then they each see 1 roommate with green nostrils and 1 with purple nostrils. This is important. Any given roommate on this day knows that if his nostrils are purple then his roommates both see 1 set of green and 1 set of purple nostrils. If that is the case, those two will kill themselves that night, knowing that both of them have green nostrils.

3 - Next day, everyone wakes up. They have each gone through the previous 2 steps, and can now come to no other conclusion than that all of them have green nostrils. If all of them did not have green nostrils, then there is no way they all could've woken up on the third day, as seen by the logic followed in the previous 2 steps.

If you enjoy logic puzzles like this, keep pluggin away at it! I was alot of fun for me. Keep in mind that this is a straight up logic/math problem, not a riddle. Also, 'They all kill themselves on the 3rd day' is the correct answer, period.

Believe it or not, figuring the first part out is actually the easier half of this problem. The hard part is figuring out exactly what information the pizza man gave them that allowed them to deduce their own nostril color. And don't ask about that...I don't fully understand it myself :p

Sunderhorn
09-26-2008, 10:10 AM
Satrina, my understanding of the Blue-Eyed Islander problem is that there doesn't even have to be any brown-eyed islanders, only the knowledge that any given islanders eyes can be either blue or brown.

The common knowledge provided allows the induction to start at n=1 for any individual...without it n=0 is the start which gets you nowhere. Right?

Satrina
09-26-2008, 10:14 AM
Ah yep, I see it now. I missed the purple/green point in step two and then it all falls into place.

I still maintain that the complete honestly clause dooms it from the start!

Alent
09-26-2008, 10:18 AM
Ah yep, I see it now. I missed the purple/green point in step two and then it all falls into place.

I still maintain that the complete honestly clause dooms it from the start!

But... but... there's no reflective surfaces and thus no ~light or color~?

Sunderhorn
09-26-2008, 10:26 AM
Alent....you're forgetting about the marmot...

Satrina
09-26-2008, 10:27 AM
Satrina, my understanding of the Blue-Eyed Islander problem is that there doesn't even have to be any brown-eyed islanders, only the knowledge that any given islanders eyes can be either blue or brown.

The common knowledge provided allows the induction to start at n=1 for any individual...without it n=0 is the start which gets you nowhere. Right?

That's correct. There can be zero brown eyed islanders and the iteration will continue the same until everyone eventually concludes that their assumption that they have brown eyes is false.

Leeroi
09-26-2008, 10:41 AM
WOOT! I got it! It took longer than it should have, but here is the answer. I start by quoting myself (isn't that frowned upon, socially?!?):


The following logic applies to any given roommate.

To stay with the assignments given by Gwydolin, C is any given roommate and A and B are the "other" two roommates:

(1) C knows that the criteria of "at least one" is met regardless of what his own color is, since A and B both have green nostrils. Thus C does not know what his color is and will not kill himself.
(2) C knows that the "at least one" criteria is met for A since A will see that B has green nostrils, and the criteria is met for A regardless of C's color. Thus C knows A will not kill himself and C knows nothing of his own color.
(3) C knows that the "at least one" criteria is met for B since B will see that A has green nostrils, and the criteria is met for B regardless of C's color. Thus C knows that B will not kill himself and C knows nothing of his own color.

So: C does not die and C knows that neither A nor B will die. C also knows that his own color is irrelevant to the fate of any of the roommates - all will continue to live because C can see that both A and B have green nostrils.

Since the "C" perspective is true for all roommates, all roommates will (1) not kill themselves and (2) know that their roommates will not kill themselves either, and (3) that his own color makes no difference in this outcome. So... no one dies.

My previous logic is correct for Day 1 only - bummer for the roommates. The answer lies in what each roommate is able to deduce from the other roommates living or dying on each successive day. This involves a lot of "I know that he knows that I know he knows..." type reasoning so you have to work slowly through the logic. OK - here comes the mind-bending part of it....

Day 1 is when the pizza boy makes his comment.

On Day 2: C wakes up and sees that both roommates (A and B) are living still, as C knew they would be.

On Day 3: C wakes up and sees that both A and B are living still - but then realize that this means C must also have green nostrils since if C had purple nostrils A and B would have killed themselves last night (Day 2). C orders pizza, beats the tar out of the Pizza Marmot, tells the marmot the color of the marmots own nostrils, and then C kills himself at midnight since C is now sure C has green nostrils, just like A and B. As an added vengeance, C also informs the marmot that his eyes are yellow....

The reason C knows he has green nostrils is because had C purple nostrils, A and B could not have arrived at the same logic C had - each of the roommates needs 2 roommates with green nostrils for them to have lived this long. Since they ALL need 2 green-nostriled roommates to have lived this long, they must ALL have green nostrils.

Looking at it differently:
If C had, in fact, had purple nostrils, A and B would both have figured out they had green nostrils and killed themselves on Day 2. This is true because C knows that A would have looked at B and C and seen that the "at least one of you" was B. C recognizes that A realizes that this means B, in order to be uncertain about his own nostrils on Day 1, would have to see green nostrils on A to continue living past Day 1 - if both A and C had purple nostrils then B would have known he was the green guy and killed himself on Day 1. That is, C knows that if C had purple nostrils, A would realize that the only way B could live past Day 1 was if A had green nostrils. Since B did live past Day 1, A would know that A had green nostrils and would have killed himself on Day 2 - unless, and only unless, C ALSO had green nostrils which would generate the uncertainty for A and allowed A to live to Day 3. Since no one died on Day 1 or Day 2, everyone will realize that they all have green nostrils and kill themselves on Day 3. Remember, all the above is from C's perspective (i.e. C knows that A would figure out that A has green nostrils because B did not die on Day 1), which is true for each roommate's perspective.

The shortest way to explain this is: The only way all three roommates live to Day 3, with each roommate knowing that the other two roommates have green nostrils, is if all three roommates have green nostrils. Since nothing else is possible, all three roommates deduce the color of their own nostrils on Day 3 and kill themselves at midnight.

More insidious is that each roommate KNOWS right away that no one will kill themselves on Day 1 and that no one will live past Day 3. If A and B kill themselves on Day 2, C knows he has purple nostrils and kills himself on Day 3. If A and B do not kill themselves on Day 2, then C knows he has green nostrils and kills himself on Day 3 anyway - but has company beating the tar out of the evil marmot.

Crap - I wasted 2 hours at work reasoning this through. But it was fun in a perverse sort of way....
EDIT: Double Crap - in the 2 hours it took me to figure it out and type out my answer, it had already been solved and explained by several others.... /Sigh

Satrina
09-26-2008, 10:53 AM
It's good for your brain.

Optimoos
09-26-2008, 11:13 AM
Wall of text

Hey Leeroi, all of that figuring seems to indicate you've got higher than 74 intellect as indicated in your sig :D

PS - Cider, wtb tankspot sigs again

Kazeyonoma
09-26-2008, 11:35 AM
me and my 3 co workers sat through this for 4 hours yesterday and today figuring it out. And most of the time spent was getting confused by 2 people talking at once LOL.

Arrivan
09-26-2008, 05:18 PM
Ugh, I'm an engineer, not a mathematician or whatever field this falls under. :(

Gwyndolin
10-24-2008, 11:32 AM
It seems that there was still some confusion to the solution. Sorry it took so long to get back to this, I lost track of the thread and forgot about it :(

If you refer to how I did my solution on page 1 I had C assume he was Purple and didn't explain why that worked very well. Basically on Day 2 nobody is dead yet, this is the point when C can figure out what color he is. He does a proof by contradiction in which he realizes that since he knows both A and B are green that it is impossible for them to still be alive if he is purple, therefor he has to be green.