[Guest]

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?

[Guest]

Since you complimented my riddle so nicely, I will do what I said I wouldn’t and respond one more time. First of all, you have again ignored my questions and merely restated your argument. It’s not that I demand to be answered or anything, but I think if you tried to answer my questions, you would realize that you could not do so and would stop restating the same argument.

So, I will offer my question in a more structured way. Suppose there are 40 people on an island: You and 39 others. Suppose you see 25 Marked people and 14 Unmarked people. Suppose further that, like the original riddle, if anyone ever learns their status, they will kill themselves at the next sunrise (and everyone will learn about the death later that day). Let’s ignore the Mercifulist complication…let’s say they are all Traditionalists.

According to your checklist, you know everything you need to deduce you state:

a. How many total people there are on the island -- FORTY

b. How many Marked people there are on the island other then me TWENTY-FIVE

c. How many Unmarked people there are on the island other then me FOURTEEN

d. That everyone on the island is aware of the existence of the same group of people – either marked or unmarked. OBVIOUSLY, THERE ARE AT LEAST 14 OF THE SMALLER GROUP

i. Note it does not matter if the islanders are aware of the majority or the minority just as long as all of them are aware of the same group.

e. That everyone on the island is aware that everyone else is aware of b c and d.

AGAIN, OBVIOUSLY, YOU KNOW THAT EVERYONE SEES AT LEAST 13 OF THE SMALLER GROUP AND THUS KNOWS THAT EVERYONE ELSE SEES AT LEAST 12

f. That there is a consequence for knowing my own state – i.e. leaving the island or suicide. (This is a given at the start of the puzzle)

YES, ON THIS ISLAND, YOU KILL YOURSELF IF YOU DEDUCE YOUR STATUS.

So, here are my questions to you: ARE YOU MARKED OR UNMARKED? WHEN WOULD YOU DIE?

By the way, it would be circular to answer to this question by simply saying that you will learn your status depending on whether 14 people kill themselves on the fourteenth day, unless you can explain how the logic of why the first person to kill him/herself could deduce his/her status. Note that there are only three possible combinations one could see: 13/26, 14/25, 15/24. So, if your answer for how the 14/25 people (you) learn their status depends on what others do (without explaining how you would know) this is only non-circular if you can explain why one of the other two possible groups could deduce their status.

Also, if you use a term like “the fourteenth day,” make sure and specify the fourteenth day AFTER WHAT?

If you can’t answer this question, that is because your checklist does not define the sufficient conditions for deducing one’s state, and you are missing the importance of what the outsider says.

And I applaud your puzzle. It has caused me to question the premise of the blue/brown puzzle and look at things differently. Even if I am ultimately proven wrong here, I will still have learned something. Your puzzle has seized my imagination for several days now, and for that I thank you.

My contention is that the statement of the Scientist does nothing to alter the state of the island. My logic is as follows:

(Sorry for the code box, I wasn't able to get the script for the lists to work correctly)

1) In order to deduce my own state I need to know the following

      a. How many total people there are on the island

      b. How many Marked people there are on the island other then me

      c. How many Unmarked people there are on the island other then me

      d. That everyone on the island is aware of the existence of the same group of people – either marked or unmarked.

           i. Note it does not matter if the islanders are aware of the majority or the minority just as long as all of them are aware of the same  group.

      e. That everyone on the island is aware that everyone else is aware of b c  and d. 

      f. That there is a consequence for knowing my own state – i.e. leaving the island or suicide. (This is a given at the start of the puzzle)


  2) Once all of these conditions are satisfied, I can deduce my own state. 

      a.  In the classic puzzle this occurs after N+1 days.


  3) As I proved above, once any group is greater then or equal to 5, I satisfy the conditions set forth in 1.


  4) Because there are 40 islanders, one group must have >=5 members in it. 

      a. Because of 3, all suicides that would have happened on the island must have happened before the Scientist ever got there.

[Guest]

What do you think of the following two changes (to fix the common knowledge problem (without drawing too much attention to recursive reasoning) and to force a correct solution to make all possible deductions):

"Their rationality is common knowledge on the island: everyone knows that the other inhabitants are equally rational, everyone knows that everyone knows it, etc. "

AND (the revised last few paragraphs):

Weeks later, the anthropologist receives an important assignment: to determine the largest and the smallest of the four possible groups (Marked Traditionalists, Unmarked Traditionalists, Marked Mercifulists, and Unmarked Mercifulists), and to determine the respective number of islanders in the largest and smallest groups.

So, on the afternoon of the day exactly three weeks from the date of his first visit, the anthropologist 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 information you asked for, but only at a terrible cost.”

What did he mean? Which of the four possible groups was largest (or, in the case of a tie, “were largest”)? How many of the 40 islanders belonged to this/these group(s)? Which of the four possible groups had the least members? How many of the 40 islanders belonged to it/them?

I agree that I should have mentioned the scenario where there are no unmarked mercifulists. There was a lot of ground to cover in the solution, and this slipped through the cracks.

I love the twist of the mercifulists and how their death will break the "common knowledge" that everyone else with the same trait had. Again, this isn't about knowing that everyone knows someone has a mark or someone doesn't have a mark. It's all about knowing what everyone else knows about what everyone knows, and as soon as a mercifulist dies, no one can be absolutely sure of what everyone knows about what everyone knows anymore. Brilliant problem.

[Guest]

I think I understand how the Common Knowledge works. Great riddle by the way.

A, B, C, D, and E are the 5 Marked people. Each Marked person (take A for instance) knows that all the Unmarked people also see the same 4 Marked people that A sees. It doesn't matter what they see on the back of A's neck. A also knows that each of the Marked people he sees can see at least 3 other Marked people. Again it doesn't matter what they see of A. So A knows that Everyone in the village can see at least 3 Marked people. A, B, C, D, and E all have this individual knowledge.

So for the iterative steps. A knows that B (seeing C,D,E) knows that C can also see D,E. Therefore A knows that B knows that C (seeing D,E) knows that D can also see E. But A doesn't know that B knows that C knows that D knows that E can see any Marked people. The question is not what does B know about C, but what does B know about C within the mind of A. From A's perspective, B doesn't know that C can see at least three Marked people. A knows that C can see 3 Marked people, but A doesn't know that B knows that C can see 3 Marked people. So it isn't Common Knowledge that Everyone can see at least 3 Marked people, even though everyone individually knows that everyone can see at least 3 Marked people.

For breaking the Common Knowledge with a Mercifulist suicide, if A commits suicide then a Marked person died. B doesn't know that C knows that D knows that E knows there are Marked people, because at that last link in the chain E knows that A is dead and thus is unsure whether any other Marked people exist.

[Guest]

Sounds right.

I think I understand how the Common Knowledge works. Great riddle by the way.

A, B, C, D, and E are the 5 Marked people. Each Marked person (take A for instance) knows that all the Unmarked people also see the same 4 Marked people that A sees. It doesn't matter what they see on the back of A's neck. A also knows that each of the Marked people he sees can see at least 3 other Marked people. Again it doesn't matter what they see of A. So A knows that Everyone in the village can see at least 3 Marked people. A, B, C, D, and E all have this individual knowledge.

So for the iterative steps. A knows that B (seeing C,D,E) knows that C can also see D,E. Therefore A knows that B knows that C (seeing D,E) knows that D can also see E. But A doesn't know that B knows that C knows that D knows that E can see any Marked people. The question is not what does B know about C, but what does B know about C within the mind of A. From A's perspective, B doesn't know that C can see at least three Marked people. A knows that C can see 3 Marked people, but A doesn't know that B knows that C can see 3 Marked people. So it isn't Common Knowledge that Everyone can see at least 3 Marked people, even though everyone individually knows that everyone can see at least 3 Marked people.

For breaking the Common Knowledge with a Mercifulist suicide, if A commits suicide then a Marked person died. B doesn't know that C knows that D knows that E knows there are Marked people, because at that last link in the chain E knows that A is dead and thus is unsure whether any other Marked people exist.

[Guest]

Okay I think I finally see where we’ve been missing each other. First, an apology: I was restating myself not to be circular, or even stubborn, but because I wasn’t sure I was being clear. Again, thank you for your patience with me. Now for your answer:

The implied third piece of information that the Scientist imparts to the group is momentum He gives the shove that starts the first domino falling and sets the whole works into motion. Or if you prefer his words act as the starter’s pistol – loudly and decisively declaring that what must happen will now begin.

To use one more metaphor, if you picture the island at the very start of the puzzle as a precisely balanced object, the Scientist acts as a weight on one side that causes the entire structure to come crashing down. This works if you view the beginning of the puzzle as a snapshot in time, everything in place just waiting for that outside force to start things up.

Moment 0 – Everything is in balance, waiting

Moment 1 – The Scientist spills the beans

Moment 2 – islanders instantly know all the information needed to deduce their status.

Your question is, essentially, if the Scientist isn’t there to give this momentum, then how do you ever start the chain? My question in turn is, how did the islanders get there in the first place?

The way I see it there are only two ways: organic growth and sudden arrival. Let us take each in turn.

Organic growth

Whether the islanders are transported one by one or if the first two started a family that grew into the island as we presently know it (the start of the puzzle), the idea here is that the island’s population grows incrementally over time.

In the beginning this is just fine, as the island grows from one to two to three, etc. As long as we never reach that magic number of 5 in either subpopulation all is well and life is good for our islanders. The instant that 5th marked child is born, however, the momentum begins.

Remember, our islanders are perfect logicians. The very instant this 5th marked person arrives on the scene all of the islanders (including our newest arrival) become instantly aware that they are now able to deduce their fate. This is the functional equivalent of the Scientist telling them all at the same time – the 5th man is the push.

Moment 0 – No sub population >=5 exists

Moment 1 – The 5th person arrives

Moment 2 – All islanders instantly know all the information needed to deduce their status.

Sudden Arrival

Whether through happenstance, alien intervention, or an act of God – all of our Islanders arrive on the island at exactly the same time. That act itself is the only impetus needed. Again, because our islanders are perfect logicians, they chain starts the moment after they all arrive.

Moment 0 – no islanders are present

Moment 1 – all islanders are present

Moment2 – islanders instantly know all the information needed to deduce their status.

Both of these scenarios are the functional equivalent to the Scientist announcing the information, and neither requires his presence.

So my answer to your question is, 14 or 15 days after Moment 1 (either Organic or Sudden). And my argument restated is: your premise cannot be, because by the rules you set up we can never get to a state where there are 40 islanders are alive in perfect balance waiting for the Scientist to tip them over the edge.

Since you complimented my riddle so nicely, I will do what I said I wouldn’t and respond one more time. First of all, you have again ignored my questions and merely restated your argument. It’s not that I demand to be answered or anything, but I think if you tried to answer my questions, you would realize that you could not do so and would stop restating the same argument.

So, I will offer my question in a more structured way. Suppose there are 40 people on an island: You and 39 others. Suppose you see 25 Marked people and 14 Unmarked people. Suppose further that, like the original riddle, if anyone ever learns their status, they will kill themselves at the next sunrise (and everyone will learn about the death later that day). Let’s ignore the Mercifulist complication…let’s say they are all Traditionalists.

According to your checklist, you know everything you need to deduce you state:

a. How many total people there are on the island -- FORTY

b. How many Marked people there are on the island other then me TWENTY-FIVE

c. How many Unmarked people there are on the island other then me FOURTEEN

d. That everyone on the island is aware of the existence of the same group of people – either marked or unmarked. OBVIOUSLY, THERE ARE AT LEAST 14 OF THE SMALLER GROUP

i. Note it does not matter if the islanders are aware of the majority or the minority just as long as all of them are aware of the same group.

e. That everyone on the island is aware that everyone else is aware of b c and d.

AGAIN, OBVIOUSLY, YOU KNOW THAT EVERYONE SEES AT LEAST 13 OF THE SMALLER GROUP AND THUS KNOWS THAT EVERYONE ELSE SEES AT LEAST 12

f. That there is a consequence for knowing my own state – i.e. leaving the island or suicide. (This is a given at the start of the puzzle)

YES, ON THIS ISLAND, YOU KILL YOURSELF IF YOU DEDUCE YOUR STATUS.

So, here are my questions to you: ARE YOU MARKED OR UNMARKED? WHEN WOULD YOU DIE?

By the way, it would be circular to answer to this question by simply saying that you will learn your status depending on whether 14 people kill themselves on the fourteenth day, unless you can explain how the logic of why the first person to kill him/herself could deduce his/her status. Note that there are only three possible combinations one could see: 13/26, 14/25, 15/24. So, if your answer for how the 14/25 people (you) learn their status depends on what others do (without explaining how you would know) this is only non-circular if you can explain why one of the other two possible groups could deduce their status.

Also, if you use a term like “the fourteenth day,” make sure and specify the fourteenth day AFTER WHAT?

If you can’t answer this question, that is because your checklist does not define the sufficient conditions for deducing one’s state, and you are missing the importance of what the outsider says.

Edited August 13, 2010 by norraist

[Guest]

But A doesn't know that B knows that C knows that D knows that E can see any Marked people

Yes A does know that B knows that C knows. For a 100% fact A knows this. A can bet the farm that B knows that C knows that D knows that E knows.

Once you reach 5 marked men, A knows absolutely without a doubt that everyone on the island both individually knows and knows that everyone knows (that everyone knows that everyone knows, ad ifinitum).

Please, read the post where I spell out why this *must* be true.

(Hint it doesn't matter if A knows how many B thinks that C thinks that D thinks that E sees -- just that he knows that D must know that E sees at least one)

(also note I'm not addressing the Merciful twist, yet.)

I think I understand how the Common Knowledge works. Great riddle by the way.

A, B, C, D, and E are the 5 Marked people. Each Marked person (take A for instance) knows that all the Unmarked people also see the same 4 Marked people that A sees. It doesn't matter what they see on the back of A's neck. A also knows that each of the Marked people he sees can see at least 3 other Marked people. Again it doesn't matter what they see of A. So A knows that Everyone in the village can see at least 3 Marked people. A, B, C, D, and E all have this individual knowledge.

So for the iterative steps. A knows that B (seeing C,D,E) knows that C can also see D,E. Therefore A knows that B knows that C (seeing D,E) knows that D can also see E. But A doesn't know that B knows that C knows that D knows that E can see any Marked people. The question is not what does B know about C, but what does B know about C within the mind of A. From A's perspective, B doesn't know that C can see at least three Marked people. A knows that C can see 3 Marked people, but A doesn't know that B knows that C can see 3 Marked people. So it isn't Common Knowledge that Everyone can see at least 3 Marked people, even though everyone individually knows that everyone can see at least 3 Marked people.

For breaking the Common Knowledge with a Mercifulist suicide, if A commits suicide then a Marked person died. B doesn't know that C knows that D knows that E knows there are Marked people, because at that last link in the chain E knows that A is dead and thus is unsure whether any other Marked people exist.

[Guest]

What do you think of the following two changes (to fix the common knowledge problem (without drawing too much attention to recursive reasoning) and to force a correct solution to make all possible deductions):

"Their rationality is common knowledge on the island: everyone knows that the other inhabitants are equally rational, everyone knows that everyone knows it, etc. "

AND (the revised last few paragraphs):

Weeks later, the anthropologist receives an important assignment: to determine the largest and the smallest of the four possible groups (Marked Traditionalists, Unmarked Traditionalists, Marked Mercifulists, and Unmarked Mercifulists), and to determine the respective number of islanders in the largest and smallest groups.

So, on the afternoon of the day exactly three weeks from the date of his first visit, the anthropologist 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 information you asked for, but only at a terrible cost.”

What did he mean? Which of the four possible groups was largest (or, in the case of a tie, “were largest”)? How many of the 40 islanders belonged to this/these group(s)? Which of the four possible groups had the least members? How many of the 40 islanders belonged to it/them?

You can find the largest and smallest groups and their numbers. But you should also ask what, if anything, you know about the numbers of the other two groups. I also think it's fair to ask what Mike's traits are.

[Guest]

Okay I think I finally see where we’ve been missing each other. First, an apology: I was restating myself not to be circular, or even stubborn, but because I wasn’t sure I was being clear. Again, thank you for your patience with me. Now for your answer:

The implied third piece of information that the Scientist imparts to the group is momentum He gives the shove that starts the first domino falling and sets the whole works into motion. Or if you prefer his words act as the starter’s pistol – loudly and decisively declaring that what must happen will now begin.

To use one more metaphor, if you picture the island at the very start of the puzzle as a precisely balanced object, the Scientist acts as a weight on one side that causes the entire structure to come crashing down. This works if you view the beginning of the puzzle as a snapshot in time, everything in place just waiting for that outside force to start things up.

Moment 0 – Everything is in balance, waiting

Moment 1 – The Scientist spills the beans

Moment 2 – islanders instantly know all the information needed to deduce their status.

Your question is, essentially, if the Scientist isn’t there to give this momentum, then how do you ever start the chain? My question in turn is, how did the islanders get there in the first place?

The way I see it there are only two ways: organic growth and sudden arrival. Let us take each in turn.

Organic growth

Whether the islanders are transported one by one or if the first two started a family that grew into the island as we presently know it (the start of the puzzle), the idea here is that the island’s population grows incrementally over time.

In the beginning this is just fine, as the island grows from one to two to three, etc. As long as we never reach that magic number of 5 in either subpopulation all is well and life is good for our islanders. The instant that 5th marked child is born, however, the momentum begins.

Remember, our islanders are perfect logicians. The very instant this 5th marked person arrives on the scene all of the islanders (including our newest arrival) become instantly aware that they are now able to deduce their fate. This is the functional equivalent of the Scientist telling them all at the same time – the 5th man is the push.

Moment 0 – No sub population >=5 exists

Moment 1 – The 5th person arrives

Moment 2 – All islanders instantly know all the information needed to deduce their status.

Sudden Arrival

Whether through happenstance, alien intervention, or an act of God – all of our Islanders arrive on the island at exactly the same time. That act itself is the only impetus needed. Again, because our islanders are perfect logicians, they chain starts the moment after they all arrive.

Moment 0 – no islanders are present

Moment 1 – all islanders are present

Moment2 – islanders instantly know all the information needed to deduce their status.

Both of these scenarios are the functional equivalent to the Scientist announcing the information, and neither requires his presence.

So my answer to your question is, 14 or 15 days after Moment 1 (either Organic or Sudden). And my argument restated is: your premise cannot be, because by the rules you set up we can never get to a state where there are 40 islanders are alive in perfect balance waiting for the Scientist to tip them over the edge.

I will try to approach my understanding of the conditions necessary to cause deaths from a different angle.

Let's assume it's the morning of August 11th and I'm a marked islander and there are a total of 10 marked islanders. I know everyone sees at least 8 marked islanders. But is there "common knowledge" about the numbers of marked islanders? No! From my perspective, I don't know if there are 9 or 10. From anyone else's perspective, I don't know if they see 8, 9 or 10 marks, and therefore without common knowledge about the numbers, I can't know whether I have a mark or not. Therefore I won't kill myself this morning. There are a number of things that can change that.

The simplest would be if the anthropologist had said on August 10th, he sees 10 marks! I can safely kill myself. I know I see 9 marks, I know exactly who sees 9 marks and I know exactly who sees 10 marks and I know exactly what everyone else knows about what everyone knows...about their mark.

But, let's assume that on the 1st, the anthropologist said he saw at least 1 mark. He hasn't yet given anyone the common knowledge they need to determine their mark. But, he has provided one critical piece of common knowledge to the group. It is now common knowledge that if there was only one marked islander, he would kill himself on August 2nd! I know everyone knows that knows that everyone knows...that there isn't only one marked islander, but that's not important. What is important, is that there is common knowledge of the seemingly simple fact, that if there was one marked islander, he would kill himself on August 2nd.

The fact that no one kill himself on August 2nd, may be a foregone conclusion, but it's still significant. It now provides everyone the common knowledge that on August 3rd, two islanders would kill themselves if there were only 2 marked islanders. This build up of common knowledge comes to a head on August 10th, when no one kills themselves. Until noon on the 10th, I didn't know if I have a mark. But now I do. I will kill myself the following morning.

This common knowledge thing is very delicate. And is why the mercifulists throw a wrench into the works. Let's say the anthropologist tells everyone he see a mark on August 1st. On August 2nd, a marked islander kill himself. I will not be able to determine if I have a mark or not. Here's why. In the previous example, everyone knows that if an islander kills himself on August 2nd, it's because he saw no marks and KNEW whether or not he had a mark (that we know this can't happen is irrelevant). Since, he killed himself for different reason breaks the slow build up of logic necessary for all the marked islanders to determine whether or not they have a mark. Traditionalists only kill themselves when they know whether they have a mark or not. Because everyone is delaying the inevitable until they KNOW, on the 10th, everyone knows. The first chain of that common knowledge is, if there were only one marked islander he would kill himself on August 2nd. We don't know that now, because he may kill himself if he's a mercifulist instead and never actually knew if he had a mark or not.

I'd also like to address your example of the sudden arrival of a marked newcomer. I had actually thought of this when trying to wrap my head around the problem. Let's say there are 9 marked islanders and now a marked new arrival appears on the island. Let's also assume that for some reason because of his arrival, the other 9 islanders decided to kill themselves (I contend this wouldn't happen). What is the new arrival supposed to do? How could he know he had a mark? How is anyone else different than the new arrival? His addition changes nothing!. The critical piece of common knowledge is still missing. What would a marked islander do tomorrow, if he was the only marked islander? We don't know, and therefore, as long as everyone avoids providing that piece of information to everyone, they are all safe.

I actually think your example of the sudden arrival makes this more clear. Let's say there are 9 marked islanders, and they see the arrival a little ways out to shore. They all know that someone with mark is about to arrive. But does this tell them that they have a mark? Certainly not. Does, this new arrival now landing on the island change anything? I doubt it. Let's look at this from the perspective of the new arrival. He is on his way up to the shore, and he see many marks. Does he now know whether or not he has a mark? Certainly not. When he lands has this changed? No.

I don't think I can explain my thoughts on this in any other way. So I will rest my case now.

[Guest]

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.

To be sure I understand what you wrote, once the anthropologist made his faux pas, the Mercifulists, knowing they will eventually know for certain, as if all were Traditionalist, that they are Marked or Unmarked, would commit suicide the next day at sunrise.

Is this interpretation correct?

If it is correct, I question the questioning that "common knowledge" is not implied. The inference in the text is that the islanders are aware of and share in a "common knowledge" that each islander is logical and rational, as it is embedded in their religion. As zealots of their religion, they would need to believe it so. If one can logically assume differently, explain.

[Guest]

To be sure I understand what you wrote, once the anthropologist made his faux pas, the Mercifulists, knowing they will eventually know for certain, as if all were Traditionalist, that they are Marked or Unmarked, would commit suicide the next day at sunrise.

Is this interpretation correct?

If it is correct, I question the questioning that "common knowledge" is not implied. The inference in the text is that the islanders are aware of and share in a "common knowledge" that each islander is logical and rational, as it is embedded in their religion. As zealots of their religion, they would need to believe it so. If one can logically assume differently, explain.

I totally see your point. But, I might have a minor quibble. Certainly, the mercifulists act as though there is common knowledge of the brilliance and rationality of the all the islanders. Only the puzzles author can make it so, by declaring it as a condition.

Certainly, it's also implied by the fact that it's not much of a puzzle if common knowledge of this fact doesn't exist. I had no issue trying to solve the puzzle without being 100% certain of the fact. I just think it's such a good puzzle, that it's worth making a simple declaration in puzzle itself to improve the puzzle, if even by just a little bit. Why not make it bullet proof?

[Guest]

I think I understand how the Common Knowledge works. Great riddle by the way.

A, B, C, D, and E are the 5 Marked people. Each Marked person (take A for instance) knows that all the Unmarked people also see the same 4 Marked people that A sees. It doesn't matter what they see on the back of A's neck. A also knows that each of the Marked people he sees can see at least 3 other Marked people. Again it doesn't matter what they see of A. So A knows that Everyone in the village can see at least 3 Marked people. A, B, C, D, and E all have this individual knowledge.

So for the iterative steps. A knows that B (seeing C,D,E) knows that C can also see D,E. Therefore A knows that B knows that C (seeing D,E) knows that D can also see E. But A doesn't know that B knows that C knows that D knows that E can see any Marked people. The question is not what does B know about C, but what does B know about C within the mind of A. From A's perspective, B doesn't know that C can see at least three Marked people. A knows that C can see 3 Marked people, but A doesn't know that B knows that C can see 3 Marked people. So it isn't Common Knowledge that Everyone can see at least 3 Marked people, even though everyone individually knows that everyone can see at least 3 Marked people.

For breaking the Common Knowledge with a Mercifulist suicide, if A commits suicide then a Marked person died. B doesn't know that C knows that D knows that E knows there are Marked people, because at that last link in the chain E knows that A is dead and thus is unsure whether any other Marked people exist.

I think I have the simplest way to show that mercifulists stop the clock for their kind. Let's look at the simplest example of this. There two marked islanders, one is a traditionalist and one is a mercifulist. The anthropologist says there is at least one islander with a mark. The next day the marked mercifulist kills himself. Now what? Nothing! The marked traditionalist has no idea why the other marked islander killed himself. The marked traditionalist could even know that the guy who just killed himself is a marked mercifulist, but he will never know why he killed himself!

Given that the clock stops in this most simple of cases, I would find it hard to be convinced that more marked people would change anything.

[Guest]

Thank you for your explanation -- like I said, up until this puzzle I never even thought about questioning the logic behind the set up and I concede I am likely wrong. I still think there is something to be explored but I'll refrain from beating a dead horse.

However I do have an...

This common knowledge thing is very delicate. And is why the mercifulists throw a wrench into the works. Let's say the anthropologist tells everyone he see a mark on August 1st. On August 2nd, a marked islander kill himself. I will not be able to determine if I have a mark or not. Here's why. In the previous example, everyone knows that if an islander kills himself on August 2nd, it's because he saw no marks and KNEW whether or not he had a mark (that we know this can't happen is irrelevant). Since, he killed himself for different reason breaks the slow build up of logic necessary for all the marked islanders to determine whether or not they have a mark. Traditionalists only kill themselves when they know whether they have a mark or not. Because everyone is delaying the inevitable until they KNOW, on the 10th, everyone knows. The first chain of that common knowledge is, if there were only one marked islander he would kill himself on August 2nd. We don't know that now, because he may kill himself if he's a mercifulist instead and never actually knew if he had a mark or not.

Doesn't this create a paradox? Merficfulist kills himself only today only if he knows for sure that on (say) Friday he will find out his status. Wouldn't he also be aware that by killing himself he would prevent exactly what he needed to be sure would happen?

I will try to approach my understanding of the conditions necessary to cause deaths from a different angle.

Let's assume it's the morning of August 11th and I'm a marked islander and there are a total of 10 marked islanders. I know everyone sees at least 8 marked islanders. But is there "common knowledge" about the numbers of marked islanders? No! From my perspective, I don't know if there are 9 or 10. From anyone else's perspective, I don't know if they see 8, 9 or 10 marks, and therefore without common knowledge about the numbers, I can't know whether I have a mark or not. Therefore I won't kill myself this morning. There are a number of things that can change that.

The simplest would be if the anthropologist had said on August 10th, he sees 10 marks! I can safely kill myself. I know I see 9 marks, I know exactly who sees 9 marks and I know exactly who sees 10 marks and I know exactly what everyone else knows about what everyone knows...about their mark.

But, let's assume that on the 1st, the anthropologist said he saw at least 1 mark. He hasn't yet given anyone the common knowledge they need to determine their mark. But, he has provided one critical piece of common knowledge to the group. It is now common knowledge that if there was only one marked islander, he would kill himself on August 2nd! I know everyone knows that knows that everyone knows...that there isn't only one marked islander, but that's not important. What is important, is that there is common knowledge of the seemingly simple fact, that if there was one marked islander, he would kill himself on August 2nd.

The fact that no one kill himself on August 2nd, may be a foregone conclusion, but it's still significant. It now provides everyone the common knowledge that on August 3rd, two islanders would kill themselves if there were only 2 marked islanders. This build up of common knowledge comes to a head on August 10th, when no one kills themselves. Until noon on the 10th, I didn't know if I have a mark. But now I do. I will kill myself the following morning.

This common knowledge thing is very delicate. And is why the mercifulists throw a wrench into the works. Let's say the anthropologist tells everyone he see a mark on August 1st. On August 2nd, a marked islander kill himself. I will not be able to determine if I have a mark or not. Here's why. In the previous example, everyone knows that if an islander kills himself on August 2nd, it's because he saw no marks and KNEW whether or not he had a mark (that we know this can't happen is irrelevant). Since, he killed himself for different reason breaks the slow build up of logic necessary for all the marked islanders to determine whether or not they have a mark. Traditionalists only kill themselves when they know whether they have a mark or not. Because everyone is delaying the inevitable until they KNOW, on the 10th, everyone knows. The first chain of that common knowledge is, if there were only one marked islander he would kill himself on August 2nd. We don't know that now, because he may kill himself if he's a mercifulist instead and never actually knew if he had a mark or not.

I'd also like to address your example of the sudden arrival of a marked newcomer. I had actually thought of this when trying to wrap my head around the problem. Let's say there are 9 marked islanders and now a marked new arrival appears on the island. Let's also assume that for some reason because of his arrival, the other 9 islanders decided to kill themselves (I contend this wouldn't happen). What is the new arrival supposed to do? How could he know he had a mark? How is anyone else different than the new arrival? His addition changes nothing!. The critical piece of common knowledge is still missing. What would a marked islander do tomorrow, if he was the only marked islander? We don't know, and therefore, as long as everyone avoids providing that piece of information to everyone, they are all safe.

I actually think your example of the sudden arrival makes this more clear. Let's say there are 9 marked islanders, and they see the arrival a little ways out to shore. They all know that someone with mark is about to arrive. But does this tell them that they have a mark? Certainly not. Does, this new arrival now landing on the island change anything? I doubt it. Let's look at this from the perspective of the new arrival. He is on his way up to the shore, and he see many marks. Does he now know whether or not he has a mark? Certainly not. When he lands has this changed? No.

I don't think I can explain my thoughts on this in any other way. So I will rest my case now.

Edited August 13, 2010 by norraist

[Guest]

However I do have an...

Doesn't this create a paradox? Merficfulist kills himself only today only if he knows for sure that on (say) Friday he will find out his status. Wouldn't he also be aware that by killing himself he would prevent exactly what he needed to be sure would happen?

The question of paradox was my question behind the question.

But perhaps it is why they are called Mercifulists. They know that if they do not 'voluntarily' die early, they will die anyway with the remainder of the island. Their choice does not guarantee the clock from stopping, though. If the islanders who committed suicide on that first day were all Unmarked (assuming the Unmarked is the larger population) the pendulum would still swing.

It leaves the question, though. How did Mike die? He could not have been a Mercifulist, as he did not die at sunrise the next day. He could not have been a lone Marked Traditionalist for the same reason. And, if there were any Marked (assuming the Marked is the smaller population or equally as large population) who died on the first day, the clock would have stopped because of the existence - or should I say extinction - of the Mercifulists, thus Mike had no reason to commit suicide.

[Guest]

Thank you for your explanation -- like I said, up until this puzzle I never even thought about questioning the logic behind the set up and I concede I am likely wrong. I still think there is something to be explored but I'll refrain from beating a dead horse.

However I do have an...

Doesn't this create a paradox? Merficfulist kills himself only today only if he knows for sure that on (say) Friday he will find out his status. Wouldn't he also be aware that by killing himself he would prevent exactly what he needed to be sure would happen?

You are right that a mercifulist would not know he would find out, given that other mercifulists could be lurking. But, the puzzle was worded very carefully: "If a Mercifulist somehow learned on Monday afternoon that – if he and everyone else followed the Traditionalist culture..." So, a mercifulist makes his determination about whether he will find out or not, assuming everyone a traditionalist. So, they can be certain, in advance, that they would find out if everyone was a traditionalist.

[Guest]

The question of paradox was my question behind the question.

But perhaps it is why they are called Mercifulists. They know that if they do not 'voluntarily' die early, they will die anyway with the remainder of the island. Their choice does not guarantee the clock from stopping, though. If the islanders who committed suicide on that first day were all Unmarked (assuming the Unmarked is the larger population) the pendulum would still swing.

It leaves the question, though. How did Mike die? He could not have been a Mercifulist, as he did not die at sunrise the next day. He could not have been a lone Marked Traditionalist for the same reason. And, if there were any Marked (assuming the Marked is the smaller population or equally as large population) who died on the first day, the clock would have stopped because of the existence - or should I say extinction - of the Mercifulists, thus Mike had no reason to commit suicide.

Mike's fate is the crucial bit of information that tells you much of what happened. There is only one way for Mike to die on day 21. There must be more than one and fewer than 19 unmarked mercifulists and there must be exactly zero marked mercifulists. The clock stops ticking on the unmarked islanders, thus allowing the marked islanders to get to day 21 before killing themselves. Sadly, the death of all the marked islanders and day 21, causes the unmarked islanders to know their status on day 21 and to cause them to kill themselves the following day.

You are right that had there been at least 1 mercifulist for both marked and unmarked islanders, much of the island would have been spared.

[Guest]

After my own faux pas in my initial reading of the problem, I wish to say I enjoyed the problem and found it clever.

As to a rewrite, I do think it needs to be made a little more clear in the explanation of the Mercifulists. The "Monday, come Friday,...as Traditionalists" was quite confusing to me as it is written. When I first read it I thought, perhaps, it was just an error in transcribing.

[Guest]

Not true. Without the anthropologist’s statement (assuming A-E are the only Marked people), A does not know this.

A says to himself:

“It’s possible that I am Unmarked. In that event, B would look around, and see only three Marked people (C, D, E). Thus, B could say to himself,

‘It’s possible that I am Unmarked. In that event, C would look around and see only two Marked people (D, E). Thus, C could say to himself,

“It’s possible that I am Unmarked. In that event, D would look around and see only one Marked person (E). Thus, D could say to himself,

‘It’s possible that I am Unmarked. In that event, E would look around and see no Marked People. Thus, I (D) don’t know whether E sees any Marked people.’

Thus, I © don’t know whether D knows that E sees any Marked people.”

Thus, I (B) don’t know whether C knows that D knows that E sees any Marked people.’

Thus, I (A) don’t know whether B knows that C knows that D knows that E sees any Marked people.”

Yes A does know that B knows that C knows. For a 100% fact A knows this. A can bet the farm that B knows that C knows that D knows that E knows.