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If you haven't seen the blue-eyed brown-eyed riddle (or one of its vartiants) on which this is based, you may want to first address how the anthropologist's comments could cause any deaths at all.

An anthropologist receives a new and prestigious assignment – studying a population of indigenous people who have lived on a secluded island for hundreds of years. They are scientifically interesting because they are perfectly rational: not only are they all brilliant, but they perfectly apply logic, such that they can answer the most complex riddles without a moment’s hesitation. They also, of course, know that the other inhabitants are equally rational.

In trying to uncover the origin of their rationality, scientists have learned several other interesting facts. According to their religion, “Marked People,” or people with a birthmark on the back of their neck (which was, at least at one time, fairly common) are considered a sort of deity. “Unmarked” people are mere mortals. One of the central tenets of their religion is to maintain equality by ensuring that no person ever finds out whether he is Marked or Unmarked. Even though everyone can see the backs of everyone else’s necks, it is forbidden to speak of one’s mark, and so no one ever finds out whether they are Marked or Unmarked.

In fact, according to their religion, if anyone ever found out whether he was Marked or Unmarked, he would be compelled to ritualistically kill him/herself at sunrise on the following morning. On midday each day, the entire population meets at their place of worship, counts heads, and gives thanks for the fact that no one was forced to commit suicide.

Some of the islanders, the “Traditionalists,” believe in suicide only once one learns whether or not he is Marked. If a Traditionalist somehow learned on Monday afternoon that, come Friday afternoon, he would find out whether or not he was Marked, he would not commit suicide until Saturday morning. The remaining islanders are “Mercifulists.” If a Mercifulist somehow learned on Monday afternoon that – if he and everyone else followed the Traditionalist culture – he would find out whether or not he was Marked on Friday afternoon, he would go ahead and commit suicide on Tuesday morning rather than delay the seemingly inevitable. Although each person either falls into the Traditionalist or Mercifulist camp, the scientists don’t know whether both views still exist or whether all the island’s inhabitants now subscribe to one or the other. The scientists also don’t know whether the islanders are aware who holds which view.

Armed with this information, the anthropologist travels to the island and attends their midday worship service. Sitting in the pew, he counts exactly 40 island natives (since everyone attends temple service, he is confident this is an accurate count), including at least a handful of both Marked and Unmarked people. After the service, he cautiously addresses the congregation, saying: “I know better than to mention to any of you whether I see a birthmark on the back of your neck or to let you know how many Marked people I see, but I think I can safely say that it is fascinating that both the Marked and Unmarked traits continue to exist after all these years…I look forward to studying your culture.”

On mid-morning of the following day, the anthropologist again arrives at the island to attend the temple service. Before reaching the temple, however, a native named Mike told him it would be better if he left. He did.

Weeks later, the anthropologist receives an important assignment: to attempt to determine the numbers of Marked Traditionalists and Marked Mercifulists on the island. So, on the afternoon of the day exactly three weeks from the date of his first visit, he makes his third trip to the island. The first inhabitant he comes across is an Unmarked woman. When he asks the woman where he could find Mike, the anthropologist learns that Mike had committed suicide. Horrified, he leaves once again.

Upon his return, he tells his boss, “I have succeeding in determining the counts you asked for, but only at a terrible cost.”

What did he mean? How many of the original 40 inhabitants were Marked Traditionalists? How many were Marked Mercifulists?

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Scenarios:

1 with birthmark (bm) - Okay, there r both types, I don't see any others, must be me. I know now. Kill myself tomorrow. Rest realize he's dead immediately. Must know now. Noone else must be marked. Therefore, I know now. Kill self in morning. All villagers dead.

2 w/bm. Okay I only see "Bob" with a BM. He hasn't killed himself in AM. THerefore there must be 2 of us with BM. Seeing I don't see anyone else having one, its gotta be me. I kill self in AM. Bob does likewise. Everyone else sees 2 with BM dead. THerefore they can't have one - so they kill themselves. All villagers dead.

3 w/bm. Okay, I see Bob and Joe with BMs. After 2 days, they are still alive so it can't be that 2 have BMs. I don't see any others so its gotta be me. Kill self in AM, Bob and Joe do likewise. Everyone else sees 3 with BMs dead, therefore they can't have one - so they kill themselves. All villagers dead.

4 w/bm. 3 days 4th day rest of village.

22 w/bm 21 days 22nd day rest of village dead.

All the villagers must have realized that they would eventually be able to determine their identity, thus the traditionalist must have immediately killed themselves the next morning. Mike was still around late morning, so he must have been a merciful variety. The woman who was asked was still alive - so she must have been a traditionalist. Mike would have killed himself that morning and the woman would have found out at the noon meeting. She would now have to kill herself in the morning. So - The only 2 we know for sure is at least 1 traditionalist, and 1 merciful type.

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In the blue/brown problem the chain starts once the information is given that the outside agent can see one of the types. "I can see a blue eyed person". This starts off the chain reaction until there have been more nights that have passed then blue eyes I can observe, I must be blue eyed.

In this case we have the outside agent stating that he can see "some of" both parties. The logic wall I'm hitting is this: Unless there are exactly 2 of either marked or unmarked (does not matter which) then no new information has been given.

1) Exactly 1/39 (marked/unmarked or vice-versa but for clarity we'll stick with marked as the minority). When the outside agent says "some" - the 1 marked know that they must be marked and will kill themselves in the morning. This would happen if they were Merciful or Traditionalist – so we know nothing further about his makeup

1a) The merciful unmarked would know that either they are marked too, and will find out the next morning (if the 1 marked does not kill himself) or that they are unmarked (when the 1 marked does kill himself). Since they will know for sure either way they will kill themselves in the morning.

1b) The rest of the tribe will kill themselves the day after, as they will have been confirmed unmarked under the same logic as 1a.

2) Exactly 2/38 – The outside agent says “some” of each – the 2 can see only 1 other marked. They instantly know that they must also be marked and will then kill themselves the next morning. No information is gained if these two were Merciful or Traditionalists.

2a) Unmarked – Merciful: I see 2 marked and know that if both die in the morning, then I must be unmarked and I will know for sure in the am. The other alternative is that both live (we know this not to be the case) the next day which would make myself a marked. (see 3b for logic) Either way, I will know for sure in the morning. So rather then wait, I will suicide right away.

2b) Unmarked – Traditionalists will suicide either the morning after the first deaths, having been confirmed as unmarked. (The unmarked suicide’s covered by the logic above)

3) 3/37 (and up) – The outside agent says “some”. Each person sees multiple of both types. No new information is gained. Everyone on the island already knows that there are some of both. And if the agent’s “information” was enough to trip everyone over the edge (detailed below) then the scientist should have discovered a village full of corpses.

3a) Marked Merciful – Suicide on day 1 under the logic of 2a.

3b) Marked Traditionalist – Suicide on day 2. – I see 2 other marks and expect them both to die under the logic from 2. When they don’t I know I must be the third and suicide the next day. If one lives and one dies – I suicide the next day anyway because I understand the logic of 3a that caused the one death and know that I must be Marked.

3c) Unmarked Merciful – Suicide on day 1 - I can see all three marked. I know that they will figure out which is which under 3a and b. The other alternative is for all three to live (we know this to not be the case) then I must also be marked. Either way I will know for sure in a few days, so I will suicide now.

3d) Unmarked Traditionalist - Suicide on day 3 – once all the marked I can see have killed themselves, I know that I am unmarked for sure.

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Oooh,,

What he said proved that the people were not all the same - that there was variation. This gave the people a slight clue as to what they were.

I'm not sure if this is helpful, and I don't know where to go from here, but...

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We know that everyone knows who has a mark on their neck, including the anthropologist. We know that the woman in the end does not.

Mike does have a mark on his neck and no-one else does. I derive that by the fact that after the anthropologist is told that Mike committed suicide, he did not need to see any other person and the only other person that he did see was the woman who hasn't a mark.

Now for the traditionalist and the mercifulist. Mike was a traditionalist - he did not kill himself the following morning, but warned that the anthropologist should not enter the camp. He will kill himself later in the week.

All others are mercifulist because now that the last marked person is gone there will not be another named today or in the future - so no one else will commit suicide from the knowledge.

And so it goes...

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In the blue/brown problem the chain starts once the information is given that the outside agent can see one of the types. "I can see a blue eyed person". This starts off the chain reaction until there have been more nights that have passed then blue eyes I can observe, I must be blue eyed.

In this case we have the outside agent stating that he can see "some of" both parties. The logic wall I'm hitting is this: Unless there are exactly 2 of either marked or unmarked (does not matter which) then no new information has been given.

1) Exactly 1/39 (marked/unmarked or vice-versa but for clarity we'll stick with marked as the minority). When the outside agent says "some" - the 1 marked know that they must be marked and will kill themselves in the morning. This would happen if they were Merciful or Traditionalist – so we know nothing further about his makeup

1a) The merciful unmarked would know that either they are marked too, and will find out the next morning (if the 1 marked does not kill himself) or that they are unmarked (when the 1 marked does kill himself). Since they will know for sure either way they will kill themselves in the morning.

1b) The rest of the tribe will kill themselves the day after, as they will have been confirmed unmarked under the same logic as 1a.

2) Exactly 2/38 – The outside agent says “some” of each – the 2 can see only 1 other marked. They instantly know that they must also be marked and will then kill themselves the next morning. No information is gained if these two were Merciful or Traditionalists.

2a) Unmarked – Merciful: I see 2 marked and know that if both die in the morning, then I must be unmarked and I will know for sure in the am. The other alternative is that both live (we know this not to be the case) the next day which would make myself a marked. (see 3b for logic) Either way, I will know for sure in the morning. So rather then wait, I will suicide right away.

2b) Unmarked – Traditionalists will suicide either the morning after the first deaths, having been confirmed as unmarked. (The unmarked suicide’s covered by the logic above)

3) 3/37 (and up) – The outside agent says “some”. Each person sees multiple of both types. No new information is gained. Everyone on the island already knows that there are some of both. And if the agent’s “information” was enough to trip everyone over the edge (detailed below) then the scientist should have discovered a village full of corpses.

3a) Marked Merciful – Suicide on day 1 under the logic of 2a.

3b) Marked Traditionalist – Suicide on day 2. – I see 2 other marks and expect them both to die under the logic from 2. When they don’t I know I must be the third and suicide the next day. If one lives and one dies – I suicide the next day anyway because I understand the logic of 3a that caused the one death and know that I must be Marked.

3c) Unmarked Merciful – Suicide on day 1 - I can see all three marked. I know that they will figure out which is which under 3a and b. The other alternative is for all three to live (we know this to not be the case) then I must also be marked. Either way I will know for sure in a few days, so I will suicide now.

3d) Unmarked Traditionalist - Suicide on day 3 – once all the marked I can see have killed themselves, I know that I am unmarked for sure.

In the blue-eyed brown-eyed riddle (a simpler version of this one), the statement that someone has blue eyes conveys new information even if everyone already knew that (ie, three blue-eyed people). A, B, and C, who have blue eyes, all know that blue-eyed people exist. Further, A knows that B knows that other blue-eyed people exist, and A knows that C knows also. What A doesn't know is whether B KNOWS THAT C KNOWS that blue eyed people exist. This is why the seemingly innocuous statement adds information. This fact would not be undone by the addition of a new piece of information that "brown eyed people exist" also.

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Ok here go

my thoughts....

The people on the island are extremely rational. That we know. Their religion states that they must commit suicide the following morning if they find out they are marked or unmarked.

A trationalist will commit suicide only after being told they are or are not marked. A mercifulist will commit suicide the next morning if they are told (say Monday) that on Friday they will be told whether that are marked or unmarked. All this we know from the riddle.

From what the anthropologist said,“I know better than to mention to any of you whether I see a birthmark on the back of your neck or to let you know how many Marked people I see, but I think I can safely say that it is fascinating that both the Marked and Unmarked traits continue to exist after all these years…I look forward to studying your culture.”, we cannot deduce that anyone would kill themselves. Why? Because they are too rational.

From a mercifulist point of view this is not enough info to let me know that I will be told that I am either marked or unmarked. From a traditionalist point of view I would not commit suicide from this info either as I am not literally told whether I'm marked or unmarked.

Then why did Mike tell the anthro to go away? Well maybe they had a meeting after the anthro left. Maybe they feared he would let the cat out of the bag from what he had said, so they asked Mike to tell him to beat it. Mike never mentioned to the anthro that people had died or that people where killing themselves. There was no mention of it. Mike just said go.

Then when the anthro comes back he comes across a woman who tells him Mike killed himself. From this info we don't know if Mike is a traditionalist or a mercifulist as we don't know when he killed himself. He could have been one or the other. He could have killed himself the week when the anthro was last there or two weeks after the anthro left. We don't know.

This riddle does not give enough info to come to a conclusion as to who is a marked traditionalist and who is a marked mercifulist. Also, one has to remember that be you marked or not if you find out you must kill yourself. That is something we must keep in mind as the creator of this riddle tries to confuse you by mentioning that the woman the anthro comes across is not marked. What difference does that make? She could be marked or unmarked and she would still have to off herself if she were to find out.

Also,why does he say "of the original". I thought all had to assemble at the temple service and he counted 40 heads, so there aren't "of the original" as they are all originals including the woman the anthro meets weeks later.

These are my thoughts. Thanks.

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So far I can wrap my mind around a few scenarios, none of which creates a solution - so therefore they can't be correct. The first is that there is 1 marked person. If Mike were the only marked person, he would know immediately upon the scientist's speech, and would have committed suicide the following sunrise, regardless of whether he was a Traditionalist or a Mercifulist. Of course, everyone else would follow suit the very next morning.

Situation 2 is that there are 2 marked people, Mike and say "Bob." After the speech, they would expect the other to commit suicide the following morning. When that doesn't happen, they would have to know that they must also be marked, and would then commit suicide the very NEXT morining. This could explain why Mike asked the scientist to leave...but if this were the case, the remaining tribe members, whether they were Traditionalist or not, would know that they must be UNmarked, and would ALL kill themselves on the fourth day no matter what. So this can't be the solution either because there wouldn't be an unmarked woman left to tell the scientist that Mike killed himself.

If there were more than two (say three) marked - Mike, Bob, and Joe. Each one, would know that there are at least two marked in the tribe. Mike would expect (if he were unmarked) that the two other marked people would follow the logic in scenario 2. If they DON'T then the only possible reason is that Mike is also marked, and Joe and Bob are waiting for the other two, respectively, to follow scenario 2's logic. Once this realization is made, Mike would kill himself the following morning, regardless of being Traditionalist or Mercifulist. Joe and Bob would follow suit the following morning. Everyone else would follow suit the next day.

***A variation on scenario 2**** if there are two marked - the first one (say Mike) would expect the other (Bob) to commit suicide if he is the only one. If Mike is a Mercifulist, he would know in advance that if Bob commits suicide that he (Mike) is unmarked, and therefore would go ahead and commit suicide the following morning, regardless of whether Bob does or not. If Mike is a Traditionalist, he knows that he will know for sure the following day and will wait an additional day to kill himself.

So the KEY is to figure out mathematically how this pattern works out to still have an Umarked person left on the 21st day...now how to figure out whether that person is a traditionalist or mercifulist - IDK! yikes

Edited by will1978

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Any brilliant, rational individual knows that "many" can mean an indefinite number, the lower bound being zero and having no upper bound. Thus, this portion of his oration did not reveal anything already known. The lower bound of the use of 'both', on the other hand, has a lower bound of one for each object both is referring to. Thus, the anthropologist revealed there was at least one each Marked and Unmarked person. In order for this statement to have revealed anything new to any of the worshippers, there would have to have been only one worshipper with the distinct trait.

The midday the worshippers saw that there was a missing individual, they would be able to deduce they all were absent the distinctness - if they logically reasoned that there was no other reason for one to be absent, die or commit suicide, - and thus, if they were "Traditionalists" or "Mercifulists", would commit suicide the next morning or midday.

Mike could have been either the individual with the distinctive trait or of the zealous group that committed suicide the morning or midday following the day the individual who possessed the distinctive trait failed to attend service, thus we cannot acertain which camp Mike was a member of - a "Traditionalist" or a "Mercifulist".

We are only outrightly given that there was one suicide, but we can deduce there may have been at least two, as we are given that a 'handful' (a small, indefinite number with a lower bound being one) of 'both' camps were represented. Thus, we know that at least one zealot with the mark and one without the mark was in the congregation.

As there was one islander who was Unmarked that still inhabited the island three weeks later, we know that there was at least one islander who belonged to neither camp - or, through brilliant, rational thought, realized their mortality and ceased being one of the zealots.

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Okay. The Anthro stated that there are both 'marked' and 'un-marked' living together. This states that there is at least on 'marked'/'un-marked' man or woman in the group. Using this info I have come up with these possible scenarios.

1: One 'marked' individual (Mike). All the 'un-marked' look at each other. They see one 'marked' man. However they are unsure if they are 'marked' or not. There could be two 'marked' people in the group. No one knows.

1a: Mike (the 'marked' individual) looks at everyone else. He sees no one else with a 'mark'. That means that he must have a 'mark' as the anthro's statement assured him that at least one person had a 'mark'. He kills himself the next morning.

1b: The people awake and count heads. They see mike missing. They all know he had a 'mark'. However none of them know if they have a 'mark'. None of them kill themselves.

1c: Three weeks later the lady tells anthro mike is dead.

2: There are two or more 'marked' individuals. All the 'un-marked' look around. They count the 'marked' yet are still not sure if they are 'marked'.

2a: The 'marked' count everyone. They see other marked and are not sure if they are 'marked' or not. No one kills themselves.

This does not properly answer the question of how many are 'traditionalist' and how many are 'mercifulist'. However it does answer how many marked there were.

Edited by NotApplicable

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In the blue/brown problem, the balance is preserved by the uncertainty. I see 49 blue and 50 brown. I do not know for sure if I'm brown or blue. Once the information is given that an outside agent sees blue, the countdown can begin. If on the 49th day all of the blues I see are dead, then I know I'm brown. If they live, I am blue and will die the next day.

The problem here is the scientist is not introducing information that would tip this balance - unless there are only 1 or 2 of a type.

Take the worst case scenario of an even split 20/20. Once the scientist says that he sees some of both types, nothing changes in the balance. As a marked person I see 19 other marked and 20 non marked. Because the scientist said he had both, the countdown that happens in the blue/brown scenario above happens for both sides at the same time. Thus on day 19, I still don't know which one I should be and neither does anyone else (they see 19 and 20 as well). Because this uncertainty persists for every member of the island, everyone lives.

Further, because it is a closed system. Either the entire island lives, or everyone dies. Assuming your scientist did indeed introduce new information (for the sake of argument) On the afternoon of day 21 there should be no one left alive.

Noon Day 0 – information is given.

Morning Day 1 – All merciful people suicide (since the outcome is inevitable, why wait)

Morning Day 19 – Either all the marked people are dead this morning (meaning I’m unmarked) or all marked people are still alive (meaning I’m also marked). Either way, I now know and so does everyone else in the island.

Morning Day 20 – We all die.

Day 21 – Scientist comes back to see dead bodies.

In the blue-eyed brown-eyed riddle (a simpler version of this one), the statement that someone has blue eyes conveys new information even if everyone already knew that (ie, three blue-eyed people). A, B, and C, who have blue eyes, all know that blue-eyed people exist. Further, A knows that B knows that other blue-eyed people exist, and A knows that C knows also. What A doesn't know is whether B KNOWS THAT C KNOWS that blue eyed people exist. This is why the seemingly innocuous statement adds information. This fact would not be undone by the addition of a new piece of information that "brown eyed people exist" also.

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The anthropologist returns on the afternoon of the 21st day after supposedly granting new information. There are 40 villagers and the worst case scenario (or best case scenario, since it would take the longest for people to kill themselves) for this would be an even 20/20 split between Marked and Unmarked. A further stipulation of the worst case scenario would be that all the villagers are Traditionalists and thus would wait to kill themselves.

This would mean that the Marked Traditionalists (all 20 of them) would have killed themselves on the 20th day. The Unmarked Traditionalists (all 20 of them) would have been doing the same countdown and would also all kill themselves on the 20th day. So how is there any villager still alive on the afternoon of the 21st day?

I also find the starting point of the riddle a bit problematic. We know that the villagers have all the information needed. They Marked know there are either 19 or 20 Marked people, and they could have started a countdown whenever they wanted. If they never start a countdown then no one will ever know how many there are and they won't have to commit suicide. So the riddle makes an assumption that the anthropologist's statement, which did not impart any new information, somehow required the villagers to start the countdown.

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It seems to me the entire construct of both this puzzle and the original Brown/Blue puzzle rests upon there being no more then 3 of the minority class. Otherwise these perfect logic machines can deduce the critical information needed to start the chain reaction in the first place.

Assume: Each person is aware that it is possible only to be Marked or Unmarked, not that any exist for sure.

Proof:

1) If there is exactly one marked man: That man sees only unmarked and has no way of knowing for sure if a marked man exists or not. Outside information would be needed to set off the chain.

2) If there are exactly two marked men. Each sees the other as marked. Knows that either that man is the only one (see above) or that he is also marked. Neither man can know for sure that the other one is aware that a marked man exists for sure. Because he cannot be sure that everyone on the island is aware that both possibilities do exist, outside information is needed to set off the chain.

3) There are exactly three marked men. We’ll consider what each person knows about themselves and just as importantly what they know the others know.

Able (our avatar here – he doesn’t know how many there are nor if he’s marked or unmarked)

In this case assume Able is unmarked.

Baker – marked

Charlie – marked

Davis – Marked

All others, unmarked

Able: Can see three marked. Knows that each of the others is aware of 2 other marked men.

Baker: Can see 2 marked people. Because Baker is unaware of his own status, all he knows for sure that Charlie and Davis know is covered under #2 above. Baker cannot assume that either Charlie or Davis know about the existence of a marked man as it is possible from Baker’s perspective that we are dealing with situation #2. The same logic applies to Charlie and Davis.

4) Assume Able is marked

In this case each of the men can see three others. They each know for themselves that marked men exists. AND because each of the other men can be projected into the role of Able above, EACH man knows that EVERY OTHER man is aware of the existence of marked men.

Since each person on the island is 100% aware of the existence of at least one marked man, no outside information must be given and the countdown starts right away. Therefore there was no one left alive for the scientist to speak with at the very beginning of the story, never mind 3 weeks later.

The anthropologist returns on the afternoon of the 21st day after supposedly granting new information. There are 40 villagers and the worst case scenario (or best case scenario, since it would take the longest for people to kill themselves) for this would be an even 20/20 split between Marked and Unmarked. A further stipulation of the worst case scenario would be that all the villagers are Traditionalists and thus would wait to kill themselves.

This would mean that the Marked Traditionalists (all 20 of them) would have killed themselves on the 20th day. The Unmarked Traditionalists (all 20 of them) would have been doing the same countdown and would also all kill themselves on the 20th day. So how is there any villager still alive on the afternoon of the 21st day?

I also find the starting point of the riddle a bit problematic. We know that the villagers have all the information needed. They Marked know there are either 19 or 20 Marked people, and they could have started a countdown whenever they wanted. If they never start a countdown then no one will ever know how many there are and they won't have to commit suicide. So the riddle makes an assumption that the anthropologist's statement, which did not impart any new information, somehow required the villagers to start the countdown.

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I've been going over this dang thing for the whole day and I came to the same thought...the only way that the scientist's speech would provide new information is if there were only one marked pperson. In this case the one marked person would have previously thought that it was possible that there were NO marked people, but upon learning that there ARE both, would have to assume HE was the marked one....If this happened, he would kill himself the following morning, regardless of whether a Traditionalist or a Mercifulist. Everyone else would understand his logic and come to the conclusion that they must all be unmarked and would then kill themselves the very next morning...no one left. But if there is more than one marked person, the scientist didn't tell them anything they didn't already know "there are both kinds" .....every tribe member already knew that....so this wouldn't cause anyone to kill themselves if they weren't already doing so. I think the initial riddle is either missing some vital info, or is incorrectly worded.

The anthropologist returns on the afternoon of the 21st day after supposedly granting new information. There are 40 villagers and the worst case scenario (or best case scenario, since it would take the longest for people to kill themselves) for this would be an even 20/20 split between Marked and Unmarked. A further stipulation of the worst case scenario would be that all the villagers are Traditionalists and thus would wait to kill themselves.

This would mean that the Marked Traditionalists (all 20 of them) would have killed themselves on the 20th day. The Unmarked Traditionalists (all 20 of them) would have been doing the same countdown and would also all kill themselves on the 20th day. So how is there any villager still alive on the afternoon of the 21st day?

I also find the starting point of the riddle a bit problematic. We know that the villagers have all the information needed. They Marked know there are either 19 or 20 Marked people, and they could have started a countdown whenever they wanted. If they never start a countdown then no one will ever know how many there are and they won't have to commit suicide. So the riddle makes an assumption that the anthropologist's statement, which did not impart any new information, somehow required the villagers to start the countdown.

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Such that N is the lesser of the number of islanders who are Marked or Unmarked, (i.e., 1 ≤ N ≤ 20),

If N = 1,

on Day 0, the 1 Marked/Unmarked islander realizes he is Marked/Unmarked

on Day 1 the 1 Marked/Unmarked islander commits suicide; and

on Day 2 the remaining 39 Marked/Unmarked islanders commit suicide

If N > 1,

on the (2N-1)th day the N Marked/Unmarked islanders realize they are Marked/Unmarked

on the 2Nth day the N Marked/Unmarked islanders commit suicide; and

on the (2N+1)th day the remaining (40 - N) Marked/Unmarked islanders commit suicide

Except for where N = 1, the first islanders who commit suicide will be on an even numbered day after the anthropoligists oration, therefore when on the 21st day (an odd number of days after the oration), as there is reported a suicide (Mike's) by a yet-living Unmarked islander it can be deduced that 10 islanders had committed suicide the previous day and they were Marked. As we are not given how many of these 10 committed suicide in the morning and-or how many committed suicide in the afternoon of the previous day, we do not know how many were Mercifulists and how many were Traditionalisits. Though the anthropologist may know, we can not ascertain whether Mike died the previous day because he was Marked or whether he was an Unmarked Mercifulist who died that morning.

As it is afternoon, with at least 1 islander yet alive it can be deduced that she must be an Unmarked Traditionalist. But, as no other information is given, the quantity of Mercifulists and Traditionalists remains unknown.

Edited by Dej Mar

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Such that N is the lesser of the number of islanders who are Marked or Unmarked, (i.e., 1 ≤ N ≤ 20),

If N = 1,

on Day 0, the 1 Marked/Unmarked islander realizes he is Marked/Unmarked

on Day 1 the 1 Marked/Unmarked islander commits suicide; and

if a Marked/Unmarked Mercifulist

on Day 2 the remaining 39 Marked/Unmarked islanders commit suicide

else, as a Marked/Unmarked Traditionalist

on Day 3 the remaining 39 Marked/Unmarked islanders commit suicide

If N > 1,

on the (2N-1)th day the N Marked/Unmarked islanders realize they are Marked/Unmarked

on the 2Nth day the N Marked/Unmarked islanders commit suicide; and

if any N Marked/Unmarked islanders were Mercifulists

on the (2N+1)th day the remaining (40 - N) Marked/Unmarked islanders commit suicide

else, as all N Marked/Unmarked islanders were Traditionalists

on the (2N+2)th day the remaining (40 - N) Marked/Unmarked islanders commit suicide

Except for where N = 1, the first islanders who commit suicide will be on an even numbered day after the anthropoligists oration, therefore when on the 21st day (an odd number of days after the oration), as there is reported a suicide (Mike's) by a yet-living Unmarked islander it can be deduced that 10 islanders had committed suicide the previous day and they were Marked. As we are not given how many of these 10 committed suicide in the morning and-or how many committed suicide in the afternoon of the previous day, we do not know how many were Mercifulists and how many were Traditionalisits. Though the anthropologist may know, we can not ascertain whether Mike died the previous day because he was Marked or whether he was an Unmarked Mercifulist who died that morning.

As those that were Marked may have all been Traditionalists, we can not deduce whether the living Unmarked woman yet alive is a Mercifulist or Traditionalist. And, with no other information given, the only quantity we may be able to attest to is the number of Marked (10) and Unmarked (30).

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For all cases N>3, the chain would start instantly and all the islanders would be dead well before the puzzle starts.

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Why 2N-1? How about N+1 (One Marked person would die on the first day, 2 would die on the second, three would die on the third)?

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Even if everyone knew that there was a marked person and that everyone else was aware of that fact, without some outside information or something else changing, there would be no "countdown." Simply knowing that everyone is aware there is a marked person does not help anyone know their own status. A countdown would only start if some other fact (such as, there is at least one Marked person and one Unmarked person) was known that let you infer what your status must be from the fact that others haven't yet killed themselves.

It seems to me the entire construct of both this puzzle and the original Brown/Blue puzzle rests upon there being no more then 3 of the minority class. Otherwise these perfect logic machines can deduce the critical information needed to start the chain reaction in the first place.

Assume: Each person is aware that it is possible only to be Marked or Unmarked, not that any exist for sure.

Proof:

1) If there is exactly one marked man: That man sees only unmarked and has no way of knowing for sure if a marked man exists or not. Outside information would be needed to set off the chain.

2) If there are exactly two marked men. Each sees the other as marked. Knows that either that man is the only one (see above) or that he is also marked. Neither man can know for sure that the other one is aware that a marked man exists for sure. Because he cannot be sure that everyone on the island is aware that both possibilities do exist, outside information is needed to set off the chain.

3) There are exactly three marked men. We’ll consider what each person knows about themselves and just as importantly what they know the others know.

Able (our avatar here – he doesn’t know how many there are nor if he’s marked or unmarked)

In this case assume Able is unmarked.

Baker – marked

Charlie – marked

Davis – Marked

All others, unmarked

Able: Can see three marked. Knows that each of the others is aware of 2 other marked men.

Baker: Can see 2 marked people. Because Baker is unaware of his own status, all he knows for sure that Charlie and Davis know is covered under #2 above. Baker cannot assume that either Charlie or Davis know about the existence of a marked man as it is possible from Baker’s perspective that we are dealing with situation #2. The same logic applies to Charlie and Davis.

4) Assume Able is marked

In this case each of the men can see three others. They each know for themselves that marked men exists. AND because each of the other men can be projected into the role of Able above, EACH man knows that EVERY OTHER man is aware of the existence of marked men.

Since each person on the island is 100% aware of the existence of at least one marked man, no outside information must be given and the countdown starts right away. Therefore there was no one left alive for the scientist to speak with at the very beginning of the story, never mind 3 weeks later.

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The only thing that keeps the balance in place is that each individual cannot be sure that everyone else knows about the existence. As you stated:

What A doesn't know is whether B KNOWS THAT C KNOWS that blue eyed people exist. This is why the seemingly innocuous statement adds information.

Unless there is an error in my logic (which I am always willing to concede is possible) I believe I've proven that if there are more then 3 marked men then everyone on the island is both aware that there are marked men in existence and, crucially, everyone is aware that everyone else is also aware of this fact.

Therefore either:

a)there is no need for an outside agent to impart the information that marked men exist as everyone can deduce this information for themselves (they are perfect logicialy)

or

b) the introduction of this information has no effect because it is already known.

(For clarity, I use the phrase 'countdown' as shorthand for the logical deduction process the islanders would use to determine which eye color they were. I see N marked men, I expect them all to die in N days. If on day N they live I must be marked.)

Even if everyone knew that there was a marked person and that everyone else was aware of that fact, without some outside information or something else changing, there would be no "countdown." Simply knowing that everyone is aware there is a marked person does not help anyone know their own status. A countdown would only start if some other fact (such as, there is at least one Marked person and one Unmarked person) was known that let you infer what your status must be from the fact that others haven't yet killed themselves.

Edited by norraist

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I don't disagree with either of those statements. Everyone knows there is a marked person and everyone knows everyone else knows it. But, it is too quick to say that therefore no information is conveyed.

If there are four marked people (A, B, C, D), A knows that B knows there are marked people. Moreover, A knows that B knows that C knows there are Marked people (since A knows that even if he is Unmarked, that even if B thinks he (B) is Unmarked (which A knows is false), that B knows that even if C thinks he © is Unmarked (which A and B know is false), that C will at least see the mark on D). However, A does not know whether B knows that C knows that D knows that there are marked people (since A is thinking that if he is Unmaked, that B is thinking that if he is unmarked, than C is thinking that if he is Unmarked, than D is thinking that he sees no marked people).

As a result, the anthropologist's statement still conveys information. It means that when D does not kill himself on the first morning, A knows that if he is unmarked, that B knows that if he is unmarked, that C now knows that he must be Marked. If C doesn't kill himself on the second morning, then A knows that if he is unmarked that B now knows that he must be Marked. When B doesn't kill himself on the third morning, A knows that he is Marked. Since this same logic for A works for B, C, and D, all four will kill themselevs on the fourth morning. This is what produces the "countdown."

It takes a good bit of rumination to convince oneself that this logic works with larger and larger numbers of people, but I think it is well-accepted that it does (see discussion of the blue-eyed brown-eyed riddle).

As to your claim that even without an outside statement, people would start killing themselves, I don't see how this follows. If A sees only Unmarked people, he has no basis to conclude what his status is (since, for all he knows, everyone is Unmarked). A knows that if there are any Marked people, it must be him, but he doesn't know whether there are Marked people and so can deduce nothing.

The only thing that keeps the balance in place is that each individual cannot be sure that everyone else knows about the existence. As you stated:

Unless there is an error in my logic (which I am always willing to concede is possible) I believe I've proven that if there are more then 3 marked men then everyone on the island is both aware that there are marked men in existence and, crucially, everyone is aware that everyone else is also aware of this fact.

Therefore either:

a)there is no need for an outside agent to impart the information that marked men exist as everyone can deduce this information for themselves (they are perfect logicialy)

or

b) the introduction of this information has no effect because it is already known.

(For clarity, I use the phrase 'countdown' as shorthand for the logical deduction process the islanders would use to determine which eye color they were. I see N marked men, I expect them all to die in N days. If on day N they live I must be marked.)

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You are addressing whether the anthropologist conveys information. I still don't buy it. You refer to (A,B,C,D) who are Marked. Though the riddle does state that there at least a handful of each so a minimum of 5, using 4 should suffice.

A,B,C, and D each individually know that there are at least three Marked people. They also each individually know that each of the three Marked people they see can also see at least two other Marked people. They also know that all the Unmarked people can see at least three Marked people. Therefore everyone in the village knows that everyone else in the village can see at least two Marked people. A does know that B knows that C knows that D knows that there are Marked people, because A knows that EVERYONE in the village can see at least two Marked people. And since they are all perfectly logical and know everyone else is too, everyone in the village knows that everyone else in the village can make the same deduction I just did.

Your description of what causes the countdown is off as well. If C really did find out that he was Marked after the first morning, then he would have to kill himself the next morning. But that would throw off the count. Also I do question the same premise of the Blue Eye riddle, and didn't find any cogent explanation as to how the Guru conveyed completely new information. The only thing I can guess is that everyone had some understanding that the Guru's pronouncement would start the countdown.

But now to the second and more substantial problem. How could any villager possibly be alive on the afternoon of the 21st day? At most 20 people are Marked and 20 people are Unmarked. According to the logic of starting the countdown as soon as the anthropologist spoke, the longest anyone could survive would be until sunrise on the 20th day, and that only if all of them are Traditionalists.

If there are four marked people (A, B, C, D), A knows that B knows there are marked people. Moreover, A knows that B knows that C knows there are Marked people (since A knows that even if he is Unmarked, that even if B thinks he (B) is Unmarked (which A knows is false), that B knows that even if C thinks he © is Unmarked (which A and B know is false), that C will at least see the mark on D). However, A does not know whether B knows that C knows that D knows that there are marked people (since A is thinking that if he is Unmaked, that B is thinking that if he is unmarked, than C is thinking that if he is Unmarked, than D is thinking that he sees no marked people).

As a result, the anthropologist's statement still conveys information. It means that when D does not kill himself on the first morning, A knows that if he is unmarked, that B knows that if he is unmarked, that C now knows that he must be Marked. If C doesn't kill himself on the second morning, then A knows that if he is unmarked that B now knows that he must be Marked. When B doesn't kill himself on the third morning, A knows that he is Marked. Since this same logic for A works for B, C, and D, all four will kill themselevs on the fourth morning. This is what produces the "countdown."

It takes a good bit of rumination to convince oneself that this logic works with larger and larger numbers of people, but I think it is well-accepted that it does (see discussion of the blue-eyed brown-eyed riddle).

As to your claim that even without an outside statement, people would start killing themselves, I don't see how this follows. If A sees only Unmarked people, he has no basis to conclude what his status is (since, for all he knows, everyone is Unmarked). A knows that if there are any Marked people, it must be him, but he doesn't know whether there are Marked people and so can deduce nothing.

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As to your first problem, I'm not sure how to explain the logic better than in my last post and won't try (perhaps someone else could help).

As to the second, you astutely observed that if everyone was a Traditionalist, no one could survive passed the 20th morning. So, what does this tell you?

You are addressing whether the anthropologist conveys information. I still don't buy it. You refer to (A,B,C,D) who are Marked. Though the riddle does state that there at least a handful of each so a minimum of 5, using 4 should suffice.

A,B,C, and D each individually know that there are at least three Marked people. They also each individually know that each of the three Marked people they see can also see at least two other Marked people. They also know that all the Unmarked people can see at least three Marked people. Therefore everyone in the village knows that everyone else in the village can see at least two Marked people. A does know that B knows that C knows that D knows that there are Marked people, because A knows that EVERYONE in the village can see at least two Marked people. And since they are all perfectly logical and know everyone else is too, everyone in the village knows that everyone else in the village can make the same deduction I just did.

Your description of what causes the countdown is off as well. If C really did find out that he was Marked after the first morning, then he would have to kill himself the next morning. But that would throw off the count. Also I do question the same premise of the Blue Eye riddle, and didn't find any cogent explanation as to how the Guru conveyed completely new information. The only thing I can guess is that everyone had some understanding that the Guru's pronouncement would start the countdown.

But now to the second and more substantial problem. How could any villager possibly be alive on the afternoon of the 21st day? At most 20 people are Marked and 20 people are Unmarked. According to the logic of starting the countdown as soon as the anthropologist spoke, the longest anyone could survive would be until sunrise on the 20th day, and that only if all of them are Traditionalists.

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Just so we're clear, I understand perfectly the logic that is used by each islander to determine if they are marked or unmarked once the scientist speaks. My contention is, unless there are three or less marked men having the scientist speak is utterly unnecessary.

However, A does not know whether B knows that C knows that D knows that there are marked people (since A is thinking that if he is Unmarked, that B is thinking that if he is unmarked, than C is thinking that if he is Unmarked, than D is thinking that he sees no marked people).

I also understand what you are saying here, but there comes a point where that logic does not hold. I currently believe it to be 4, but lets call it X for arguments sake as I may be too low.

The problem is at some point another logic must also come into play.

If I know that everyone individually knows that marked people exist, then in order for the balance to still exist I must also believe that it is possible that someone else can believe that a third person is not aware of this fact. (Like you state in the quote above.)

In other words I must be able to believe that one person on the island can believe that there does not exist any marked people. If I cannot believe that, then I must conclude that every one not only knows that marked people exist but that everyone knows that everyone else knows.

Let us take it to an extreme to illustrate the point.

Assume that there are 100,000 marked and 100,000 unmarked islanders. Under your logic above you can follow the chain down from #100,000 (Well I know 99,999 knows but does he know that 99,998 knows, etc) until you get to the point where you are down to A B C and D as above.

In order for anyone to think that someone else might think that there are no marked people; this has to be at least possible. Because I can see 99,999 marked and 100,000 unmarked I know that there are only two cases that are technically possible: There is an even 100,000 balance or there are 99,999 marked and 100,001 unmarked. I know this for a fact.

Because I know this for a fact, I also know that there exists only a small amount of scenarios That any other person can conceive of. 1) that there are 99,998 marked and 100,002 unmarked. (I really am unmarked and the other guy also believes hes unmarked). 2) That there are 99,999 marked, and 100,001 unmarked (I really am marked, and the other guy

thinks that he is unmarked.) #3 and #4 are just the two cases above with the labels reversed.

Further because of this, even if I follow the logic chain you state all the way to the inevitable conclusion (99,999 thinking that hes unmarked thinking that 99,998 thinks that hes unmarked, thinking that…. 2 thinks that hes unmarked thinks that 1 thinks that there are no marked people) I know this cannot possibly be. The best case scenario that anyone can possibly be faced with is that there are 99,998 marked people.

In other words there are simply too many marked people present to allow for anyone to think that none exist. If I know that no one can possibly think that none exist, then I know that everyone else must be aware of this fact too.

Because of this, I have satisfied the two key components to starting the suicide countdown. 1) Everyone is aware of the existence of marked men. 2) Everyone is aware that everyone else must be aware of this fact.

I don't disagree with either of those statements. Everyone knows there is a marked person and everyone knows everyone else knows it. But, it is too quick to say that therefore no information is conveyed.

If there are four marked people (A, B, C, D), A knows that B knows there are marked people. Moreover, A knows that B knows that C knows there are Marked people (since A knows that even if he is Unmarked, that even if B thinks he (B) is Unmarked (which A knows is false), that B knows that even if C thinks he © is Unmarked (which A and B know is false), that C will at least see the mark on D). However, A does not know whether B knows that C knows that D knows that there are marked people (since A is thinking that if he is Unmaked, that B is thinking that if he is unmarked, than C is thinking that if he is Unmarked, than D is thinking that he sees no marked people).

As a result, the anthropologist's statement still conveys information. It means that when D does not kill himself on the first morning, A knows that if he is unmarked, that B knows that if he is unmarked, that C now knows that he must be Marked. If C doesn't kill himself on the second morning, then A knows that if he is unmarked that B now knows that he must be Marked. When B doesn't kill himself on the third morning, A knows that he is Marked. Since this same logic for A works for B, C, and D, all four will kill themselevs on the fourth morning. This is what produces the "countdown."

It takes a good bit of rumination to convince oneself that this logic works with larger and larger numbers of people, but I think it is well-accepted that it does (see discussion of the blue-eyed brown-eyed riddle).

As to your claim that even without an outside statement, people would start killing themselves, I don't see how this follows. If A sees only Unmarked people, he has no basis to conclude what his status is (since, for all he knows, everyone is Unmarked). A knows that if there are any Marked people, it must be him, but he doesn't know whether there are Marked people and so can deduce nothing.

Edited by norraist

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The anthropologists' statement did not impart knowledge to any that Marked or Unmarked islanders existed, as each could see that for themselves. What it did was inform everyone that a single Marked or Unmarked person could now ascertain he or she was one with the distinctive trait. Before the anthopologists' statement a lone Marked/Unmarked islander would think: "Though I see 39 Unmarked, I do not know if I am Marked or Unmarked." But after the oration, "I see 39 unmarked, therefore I must be marked!"

To restate - before, the logic chain that each could deduce only worked down to the one person link and not to zero, thus the chain was broken and none could ascertain whether they themselves were Marked or Unmarked. But after the byte of knowledge, each islander's logic chain was now connected down to zero permitting each to deduce whether they themselves were Marked or Unmarked by the passing of days.

As the timer was set by the anthropologist's speech, I shall try to demonstrate how each islander would deduce whether they were Marked or Unmarked. For simplicity, let us assume there were only 10 islanders, 6 Unmarked and 4 Marked.

I should amend my amended solution. The islander realizes he or she is Marked on the (2N-2)th day, not on the (2N-1)th day. This does alter my final analysis. :(

Day 1, AM: *No suicide.*

Day 1, Midday: 'Everyone is here. but then the lone Marked islander may be a Traditionalist.'

Day 1, PM: *No suicide.*

Day 2, AM: *No suicide.*

Day 2, Midday: 'Everyone is here. There is no lone Marked islander. I, one who sees one other Marked islander, must be Marked.'

Day 2, PM: *No suicide*.

Day 3, AM: *No suicide*.

Day 3, Midday: «Everyone is here. but then both Marked islanders may be Traditionalists.»

Day 3, PM: *No suicide*

Day 4, AM: *No suicide*

Day 4, Midday: «Everyone is here. There are not only two Marked islanders. I, one who sees two other Marked islanders, must be Marked.»

Day 4, PM: *No suicide*

Day 5, AM: *No suicide*

Day 5, Midday: "Everyone is here. but then the three Marked islanders may be Traditionalists."

Day 5, PM: *No suicide*

Day 6, AM: *No suicide*

Day 6, Midday: "Everyone is here. There are not only three Marked islanders. I, one who sees three other Marked islanders, must be Marked."

Day 6, PM: *No suicide*

------

Now the four Marked islanders know they are Marked. We have two conditions - a Mercifulist exists or not.

------

Version 1: At least one Mercifulist exists.

Day 7, AM: m Marked Mercifulists commit suicide.

Day 7, Midday: (The Unmarked): "I must be Unmarked as I see that there are Marked not present, thus they must have committed suicide."

Day 7, PM: 4-m Marked Traditionalists commit suicide.

Day 8, AM: n Unmarked Mercifulists commit suicide.

Day 8, Midday: "Alas, our final day."

Day 8, PM: 6-n Unmarked Traditionalists commit suicide.

------

Version 2: All are Traditionalists.

Day 7, AM: *No suicide*

Day 7, Midday: "Alas, the final day for some of us."

Day 7, PM: 4 Marked Traditionalists commit suicide.

Day 8, AM: *No suicide*

Day 8, Midday: (The Unmarked): "Alas, tomorrow we all will have died."

Day 8, PM: *No suicide*

Day 9, AM: m Unmarked Mercifulists commit suicide.

Day 9, Midday: “Death, the one appointment we all must keep, and for...who is this Charlie Chan, anyway?"

Day 9, PM: 6-m Unmarked Traditionalists commit suicide.

======

As the anthropologist arrives 21 days after his faux pas, there was a suicide and there still exists an Unmarked islander, and it being an odd day, ...

Version 1, the final day of the Marked:

11 islanders were Marked, 29 are Unmarked. Mike could have been one the Marked Mercifulists. The Marked Traditionalists and the Unmarked could still be alive. And, the Unmarked woman could be of either camp.

Or,

version 2.1, the final day of the Marked:

10 islanders were Marked, 30 are Unmarked. Mike would have been a Marked Mercifulist. The Marked Traditionalists may be off committing suicide. The Unmarked know tomorrow is their last day. The Unmarked woman may be one of either camp.

Or,

version 2.2, the final day of the Unmarked:

10 islanders were Marked, 30 were Unmarked. Mike could have been of either camp of the Marked or an Unmarked Mercifulist. Only (some of) the Unmarked Traditionalists are alive, perhaps the Unmarked woman is the last.

======

To answer the question why not N+1, it is because there are two camps, the information that no suicide provides to each islander as to how many Marked or Unmarked delays the certainty of each islander.

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let's say that of all the people (40? 100?, it doesn't matter), there are 3 Marked people: A, B, and C. As you say, everyone knows there is at least one marked person, and everyone knows that everyone knows this. No anthropologist or anyone else arrives. So, no one believes there is anyone on the island who believes that there are no marked people. Why would this lead to anyone killing themselves? How could A possibly know what's on the back of his neck (the basis of suicide)? When?

Just so we're clear, I understand perfectly the logic that is used by each islander to determine if they are marked or unmarked once the scientist speaks. My contention is, unless there are three or less marked men having the scientist speak is utterly unnecessary.

I also understand what you are saying here, but there comes a point where that logic does not hold. I currently believe it to be 4, but lets call it X for arguments sake as I may be too low.

The problem is at some point another logic must also come into play.

If I know that everyone individually knows that marked people exist, then in order for the balance to still exist I must also believe that it is possible that someone else can believe that a third person is not aware of this fact. (Like you state in the quote above.)

In other words I must be able to believe that one person on the island can believe that there does not exist any marked people. If I cannot believe that, then I must conclude that every one not only knows that marked people exist but that everyone knows that everyone else knows.

Let us take it to an extreme to illustrate the point.

Assume that there are 100,000 marked and 100,000 unmarked islanders. Under your logic above you can follow the chain down from #100,000 (We’ll I know 99,999 knows but does he know that 99,998 knows, etc) until you get to the point where you are down to A B C and D as above.

In order for anyone to think that someone else might think that there are no marked people; this has to be at least possible. Because I can see 99,999 marked and 100,000 unmarked I know that there are only two cases that are technically possible: There is an even 100,000 balance or there are 99,999 marked and 100,001 unmarked. I know this for a fact.

Because I know this for a fact, I also know that there exists only a small amount of scenarios That any other person can conceive of. 1) that there are 99,998 marked and 100,002 unmarked. (I really am unmarked and the other guy also believes he’s unmarked). 2) That there are 99,999 marked, and 100,001 unmarked (I really am marked, and the other guy

thinks that he is unmarked.) #3 and #4 are just the two cases above with the labels reversed.

Further because of this, even if I follow the logic chain you state all the way to the inevitable conclusion (99,999 thinking that he’s unmarked thinking that 99,998 thinks that he’s unmarked, thinking that…. 2 thinks that he’s unmarked thinks that 1 thinks that there are no marked people) I know this cannot possibly be. The best case scenario that anyone can possibly be faced with is that there are 99,998 marked people.

In other words there are simply too many marked people present to allow for anyone to think that none exist. If I know that no one can possibly think that none exist, then I know that everyone else must be aware of this fact too.

Because of this, I have satisfied the two key components to starting the suicide countdown. 1) Everyone is aware of the existence of marked men. 2) Everyone is aware that everyone else must be aware of this fact.

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