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   Author  Topic: BROWN EYES AND RED EYES  (Read 64739 times)
Grimbal
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Re: BROWN EYES AND RED EYES  
« Reply #225 on: May 23rd, 2013, 4:56am »
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There still is a problem:
 
If there is only one REM, he will reason that if he doesn't commit suicide, then everybody else will commit suicide on the next night.  So there will be nobody to blame him for his misconduct.
 
With 2 REMs, after the first night, each REM will consider the possibility that the other one is bluffing.  They will wait another day just to be sure.
 
And so on...
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Re: BROWN EYES AND RED EYES  
« Reply #226 on: May 24th, 2013, 3:41am »
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on May 23rd, 2013, 4:56am, Grimbal wrote:
There still is a problem:
 
If there is only one REM, he will reason that if he doesn't commit suicide, then everybody else will commit suicide on the next night.  So there will be nobody to blame him for his misconduct.
 
With 2 REMs, after the first night, each REM will consider the possibility that the other one is bluffing.  They will wait another day just to be sure.
 
And so on...

Yeah, as soon as the monks no longer trust each other to follow the rules, the deductions based on other monks following the rules break down.
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riddler358
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Re: BROWN EYES AND RED EYES  
« Reply #227 on: May 6th, 2016, 11:30pm »
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This one haunts me
 
I keep thinking so called cannon solution breaks, I think it breaks at 3 REM case
For 3 REM imagine his point of view, he expects 2 to die only because each of them expects 1 to die, and therefore if 2 don't die he concludes he have red eyes but that breaks his deduction because if he have red eyes his deduction was incorrect
 
I came up with an example (maybe not the best)
We wonder if biggest natural number N is odd or even. We assume it's odd. But we can make N+1 and it would be even. Then we can make N+2 and this would be odd. And so on. And we can go forever, but our base assuption is broken, N is no longer biggest natural number, so how can we base anything that follows on this?
 
EDIT: another example (maybe better) Willy Wutang goes on a family road trip with their camper, he likes to be prepared and have 2 spare tires, his wife sometimes gets 1 more spare tire, sometimes not, so Willy have either 2 or 3 spare tires, during first stop their kid says - dad, we have at least one spare tire, should willy panic and buy one more spare tire?
« Last Edit: May 6th, 2016, 11:44pm by riddler358 » IP Logged
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Re: BROWN EYES AND RED EYES  
« Reply #228 on: May 7th, 2016, 12:01am »
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Try drawing out the graph of possible worlds. The place people always go wrong is looking at the various what-if scenarios from the wrong perspective. When reasoning about what what a monk in a 3-REM world thinks a monk in a 2-REM world would think, you _shouldn't_ use the 3-REM world monks perspective, but the 2-REM worlds monk, because that's whose thinking you're considering.
 
The problem isn't with the canon solution. The solution is a mathematical fact; you can prove it from the axioms of modal logic. So the only problem is reconciling it with the way you think about the puzzle.
« Last Edit: May 7th, 2016, 12:04am by towr » IP Logged

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Re: BROWN EYES AND RED EYES  
« Reply #229 on: May 7th, 2016, 12:36am »
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i think i'm trying to do that, but 2 REM perspective makes it possible and proves it was all along 3 REM perspective where it's impossible
 
EDIT: for me it's like this: 1 REM is 1D world, 2 REM is 2D world, 3 REM is 3D world, one might jump from 1D to 2D in both ways, one might jump from 2D to 3D both ways, but one jump from 1D to 3D seems impossible to me, sure you can make 2 jumps, but is that the question?
« Last Edit: May 7th, 2016, 12:53am by riddler358 » IP Logged
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Re: BROWN EYES AND RED EYES   REM_again1.png
« Reply #230 on: May 7th, 2016, 2:39am »
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on May 7th, 2016, 12:36am, riddler358 wrote:
sure you can make 2 jumps, but is that the question?
Yes. Because the 3REM-world Monk (3RWM) thinks about what the 2RWM thinks about what the 1RWM thinks about the 0RWM. Which is that the latter can't exist once the public announcement is made that there is at least one REM. And that chain of beliefs cascades up.
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Re: BROWN EYES AND RED EYES   rem.png
« Reply #231 on: May 7th, 2016, 5:43am »
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on May 7th, 2016, 2:39am, towr wrote:

Yes. Because the 3REM-world Monk (3RWM) thinks about what the 2RWM thinks about what the 1RWM thinks about the 0RWM. Which is that the latter can't exist once the public announcement is made that there is at least one REM. And that chain of beliefs cascades up.

But if it turns out to be 3REM case then every one of them in the first place saw 2REM and knew each one of them saw at least 1REM, so no new information arises, and noone was to think 0REM case to begin with.
 
I see it like this
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Re: BROWN EYES AND RED EYES  
« Reply #232 on: May 7th, 2016, 9:20am »
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on May 7th, 2016, 5:43am, riddler358 wrote:

But if it turns out to be 3REM case then every one of them in the first place saw 2REM and knew each one of them saw at least 1REM, so no new information arises, and noone was to think 0REM case to begin with.
 
I see it like this

 
So, Alex, Bob and Charlie are the 3 REMs:
 
Charlie looks at Alex and Bob and thinks "I can see two REMs, so we're either in 3REM or 2REM world. If we are in 2REM world, then Bob would be looking at Alex and thinking 'I can see one REM, so we're either in 2REM world or 1REM world. If we are in 1REM world, then Alex has just found out that we aren't in 0REM world and will suicide tonight. If Alex is still around tomorrow, then we must be in 2REM world.' so Bob (and Alex) will suicide tomorrow night. If they don't, then we must be in 3REM world."
 
The key is that in the 2REM world Charlie thinks is possible, Charlie does not have red eyes. So those versions of Alex and Bob both only see 1 REM so are deciding between 2REM and 1REM worlds. The 2REM Bob is not the same person as the actual 3REM Bob
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Re: BROWN EYES AND RED EYES   REM_again2.png
« Reply #233 on: May 7th, 2016, 12:47pm »
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on May 7th, 2016, 5:43am, riddler358 wrote:
I see it like this
In this model the number of red-eyed monks is the only thing distinguishing possible worlds, so you can't have two different 2REM worlds.
You're also reasoning like the people in the 2REM world have knowledge from the 3REM world, but they don't.
Maybe the model is to simplistic.
 
So, instead consider the following starting model (attached). Each world is exactly characterized by which of Alex, Bob and Charlie is a red-eyed-monk. And the links labeled A,B,C are what they respectively consider possible.
Before the announcement all relations are symmetrical: they don't know whether they have red eyes or not, so in the world where they have red eyes, they consider it possible they don't and vice versa. But the eyes of their fellow monks are known to them, so those are the same in both worlds linked for a given monk.
 
The announcement breaks that symmetry and unravels the graph like before.
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Re: BROWN EYES AND RED EYES  
« Reply #234 on: May 8th, 2016, 12:55am »
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i read and tried to undarstand your point of view, but it's even less convincing for me now, you guys are most likely right, but i'll try to explain my point of view once again
 
- let's agree that BEM have zero influence on the situation, only REM matter
 
- from monk point of view there are 2 possible worlds in every situation: one is real and one is imaginary (we don't know which one is real) they differ only by 1 REM
 
- we all agree that in 2 REM case they die
 
- crucial case is 3 REM -> it splits into:
-> imaginary 2 REM and 1 BEM which is exactly the same as 2 REM we can remove ourselves from this case they die we agreed on that
-> real case 3 REM in which evey REM sees 2 REM he can imagine 2 REM case but he cannot hyphotesise 1 REM case, none of 3 REM can do that, it is not possible, so my conclusion here is => every monk sees 2 REM and he can't hyphotesise himself world with 1 REM, he might think such world exists in one of other monks head, but it doesn't, maybe that's the new information i'm trying to grasp, lack of imaginary world with 1 REM that reveals for him after 2 nights
 
it's still very confusing, somehow i cannot imagine 100 red eyed monks sitting there, and each one of them thinking yeah in their heads they think there might be only 1 red eyed monk that will suicide this night
 
 
 
EDIT: one other example that might be related
i insult towr => he tells grimbal i insulted him => grimbal acts => i insult tower for telling grimbal => he tells grimbal i insulted him => grimbal acts ... and so on
 
if grimbal tells me to apologize for each insult then we can roll this back
i apologize => towr accepts => grimbal asks me to apologize for previous insult => i apologize ... and so on to the first one
 
but if grimbal blocks my account we cannot roll this back unless i create new account
 
so above implication is perfectly correct but if grimbal acts blocking my account it breaks the loop and in order to insult towr again i need to create new account (new universe), we cannot go back apologizing for each insult, because i have no access to account from which i insulted
« Last Edit: May 8th, 2016, 1:09am by riddler358 » IP Logged
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Re: BROWN EYES AND RED EYES  
« Reply #235 on: May 8th, 2016, 1:18am »
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one more take on 3 REM case
 
I'm sitting there thinking: if i have brown eyes then those two guys can imagine 1 REM world
 
then after 2 nights: whoaa, they didn't suicide, therefor i have red eyes, but this means, they couldn't ever imagined 1 REM world in the first place
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Re: BROWN EYES AND RED EYES  
« Reply #236 on: May 8th, 2016, 6:11am »
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on May 8th, 2016, 1:18am, riddler358 wrote:
one more take on 3 REM case
 
I'm sitting there thinking: if i have brown eyes then those two guys can imagine 1 REM world
 
then after 2 nights: whoaa, they didn't suicide, therefor i have red eyes, but this means, they couldn't ever imagined 1 REM world in the first place

 
That's right - and the reason they couldn't have imagined 1REM world is the same as the reason why they survived the second night - that you have red eyes. If it helps, you could think of the process as trying to figure out whether they think 1REM world is possible.
 
Also, the same issue arises when there are only 2 REMs to start with - they each look at the other and wonder whether they think 0REM is possible. When the other guy survives the first night, that means they couldn't ever have imagined 0REM world in the first place, so the original announcement didn't tell them anything.
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Re: BROWN EYES AND RED EYES  
« Reply #237 on: May 8th, 2016, 11:50pm »
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yet another take on this riddle
 
I really like analogy to n-dimensional world, let us make another riddle based on that:
- we all live in 3D world or in 2D world (virtual reality), we cannot tell which one
- every person who will come to knowledge in which world he lives in needs to commit suicide and this should be announced in the morning to everyone
- some alien comes and states -> your world have at least one dimension
- what will happen?
 
in 3 REM world we all know it's either 2 REM or 3 REM, and possibility of imagining 1 REM world doesn't change that it is 2 REM world or 3 REM world, why should one act upon impossible imaginary possibility?
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Re: BROWN EYES AND RED EYES  
« Reply #238 on: May 9th, 2016, 7:45am »
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on May 8th, 2016, 11:50pm, riddler358 wrote:
yet another take on this riddle
 
I really like analogy to n-dimensional world, let us make another riddle based on that:
- we all live in 3D world or in 2D world (virtual reality), we cannot tell which one
- every person who will come to knowledge in which world he lives in needs to commit suicide and this should be announced in the morning to everyone
- some alien comes and states -> your world have at least one dimension
- what will happen?
 
in 3 REM world we all know it's either 2 REM or 3 REM, and possibility of imagining 1 REM world doesn't change that it is 2 REM world or 3 REM world, why should one act upon impossible imaginary possibility?

 
If it were actually a 2REM world, then 1REM world would be possible, which is why events there get taken into account indirectly.
 
The n-dimensional world analogy doesn't quite work because removing me from the equation doesn't turn the world into a 2D one that the inhabitants think could be 2D or 1D.
 
 
Let's try another approach:
 
Rather than starting by saying there are definitely a specific number of REMs present, which is information none of the monks has, let's start with what one of the monks does know: suppose you're a monk and you see exactly 2 REMs. What happens then?
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Re: BROWN EYES AND RED EYES  
« Reply #239 on: May 9th, 2016, 10:35am »
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Huh, I thought we were done already.
 
 
How about a different approach all together.
 
 
Base case (N=1):  
    If there is 1 REM, then he will kill himself at the end of the first day.
 
Induction hypothesis (what we want to prove for all N):  
    If there are N REMs, these kill themselves at the end of the Nth day.
 
Induction step (assuming it's true for all 1<=k<=N, then is it true for N+1?):  
    If there are N+1 REMs, then by the induction hypothesis, after the Nth day we know there were not N or fewer REMs, because they would have killed themselves.
    Therefore, on the N+1th day, the N+1 REMs
    * know the number of REMs is not any number from 1 to N (inclusive),  
    * know the number of REMs is not 0, because it was announced there's at least one and also they can see N REMs
    * will realize that because they see only N REMs, they must be the N+1th one
   and thus they will kill themselves at the end of the N+1th day.
 
Therefore by induction, from the base case and the induction step, for all N >=1 if there are N REMs, they kill themselves at the end of the Nth day.
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Re: BROWN EYES AND RED EYES  
« Reply #240 on: May 9th, 2016, 12:16pm »
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on May 9th, 2016, 7:45am, rmsgrey wrote:

Let's try another approach:
 
Rather than starting by saying there are definitely a specific number of REMs present, which is information none of the monks has, let's start with what one of the monks does know: suppose you're a monk and you see exactly 2 REMs. What happens then?

 
as i mentioned before there are 2 cases:
2 REM case -> both die because each of them can imagine that other one is imagining 0 REM case
3 REM case -> noone can imagine that any of other 2 are imagining 0 REM case to begin with
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Re: BROWN EYES AND RED EYES  
« Reply #241 on: May 9th, 2016, 12:58pm »
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on May 9th, 2016, 12:16pm, riddler358 wrote:

as i mentioned before there are 2 cases:
2 REM case -> both die because each of them can imagine that other one is imagining 0 REM case
3 REM case -> noone can imagine that any of other 2 are imagining 0 REM case to begin with
Yeah, but that's not the issue. They can imagine someone imagining someone imagining the 0 REM case.
If I'm the third REM, then I'm looking at the other two and thinking, huh, I'm sure hoping they can imagine someone considering the 0 REM case, otherwise I'm screwed. Because that would be what's going on if it were the 2REM case. The only thing I know is that either of them can see at least one person with red eyes (and I hope it's only one). But if they only see one person with red eyes, then the person they see would be considering there might be no REMs.  
Even though I know that that's not the case, I don't know whether they know that. The only way they would definitely not consider that is if I have red eyes, but since I don't know whether I have red eyes, I don't know whether they can or can't imagine anyone considering a 0 REM world. (Until after two days they're still alive.)
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Re: BROWN EYES AND RED EYES  
« Reply #242 on: May 9th, 2016, 1:12pm »
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@towr
 
i think induction is flawed in this case, because you go ahead and say
(N=1) => (N=2) => (N=3)
but if (N=3) then (N=1) is impossible
 
i say i agree on (N=1) => (N=2)
and i agree on (N=2) => (N=3)
but connection between (N=1) and (N=3) is lost
because your base case says "If there is 1 REM"
 
 
 
i think i have great example for this one
let's imagine a world where people live short lives and noone ever met their grandparents
in such world my grandfather can talk to my father
and my father can talk to me, but that doesn't mean my grandfather can talk to me, link is lost, but my father who knew both of us can imagine us talking with each other very vividly
 
 
 
PS just a thought - how about proving by induction that sum 1 + 2 + 3 + ... is a positive number?
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Re: BROWN EYES AND RED EYES  
« Reply #243 on: May 9th, 2016, 1:25pm »
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on May 9th, 2016, 12:58pm, towr wrote:

The only way they would definitely not consider that is if I have red eyes, but since I don't know whether I have red eyes, I don't know whether they can or can't imagine anyone considering a 0 REM world. (Until after two days they're still alive.)

i think that the fact that you don't know if they can consider something doesn't mean they can consider it
 
let's try different approach, convince me that in 100 REM case anyone thinks that anyone can imagine 0 REM world.  
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Re: BROWN EYES AND RED EYES  
« Reply #244 on: May 9th, 2016, 10:28pm »
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on May 9th, 2016, 1:12pm, riddler358 wrote:
@towr
i think induction is flawed in this case, because you go ahead and say
(N=1) => (N=2) => (N=3)
but if (N=3) then (N=1) is impossible
Whether 1 single REM would kill himself or not is not in any way influenced by how many REMs there are.
Sure, if there are 3 REMs, then there is not just 1 REM. 3 is not 1. We can all agree on that. And 3 is not 2 either. But that's not the issue.
 
If there is one single REM, then he would kill himself on the first night.
If N REMs kill themselves on the Nth night, then logically, N+1 REMs kill themselves on the N+1th night.
Therefore by the rule for induction, it's true for all N. There's just no wiggle room.
 
 
Quote:
i think i have great example for this one
let's imagine a world where people live short lives and noone ever met their grandparents
in such world my grandfather can talk to my father
and my father can talk to me, but that doesn't mean my grandfather can talk to me, link is lost, but my father who knew both of us can imagine us talking with each other very vividly
I don't see the relevance.
What you're suggesting is more equivalent to saying you haven't got a (by now possibly deceased) great-grandfather because you can't image talking to him. But whether you can talk to him is irrelevant. Your father's father could talk to him, therefore once upon a time he existed.
 
In this case each world with N+1 REMs can imagine (talk to) the N REM world. And therefore the 0 REM world has influence. Not because there is any direct link from N+1 REMs to 0 REMs, anymore than you can talk to your oldest ancestor.
 
 
Quote:
PS just a thought - how about proving by induction that sum 1 + 2 + 3 + ... is a positive number?
If by ... you means that it goes on forever, then to start with it's not true if we only consider natural numbers.  
But 1 + .. + N is a positive natural number for any positive natural number N
* 1 is a positive natural number
* if 1 + .. + N is a positive natural number, then 1 + .. + N + (N+1) is a positive natural number (because a positive natural number plus another positive natural number is always a positive natural number)
* therefore by induction, for any positive natural number N: 1 + .. + N is a positive natural number
 
(infinity is not a positive natural number, so it doesn't work for 1 + 2 + 3 + ... where ... means to go on forever)
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Re: BROWN EYES AND RED EYES  
« Reply #245 on: May 9th, 2016, 10:34pm »
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on May 9th, 2016, 1:25pm, riddler358 wrote:

i think that the fact that you don't know if they can consider something doesn't mean they can consider it
What matters is that they might. We're talking hypotheticals here. As long as there's doubts, I don't have to kill myself. But time will remove those doubts, because while I don't know whether they do consider it or not for the first 2 days, I know they can't consider it after the 2nd day, but until then they might, which is all the straw I need to grasp unto.
 
Quote:
let's try different approach, convince me that in 100 REM case anyone thinks that anyone can imagine 0 REM world.
Why would I try to convince you of something that's neither true nor relevant?
No one needs to think that someone can imagine a 0 REM world, what they need to consider is whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider whether anyone might consider a 0 REM world.
Which is an entirely different proposition. Because we're so deep into hypothetical worlds that the final person doesn't need to exist in the real world (or in fact the second), anymore than that your hundredth ancestor needs to still be alive for your 99th ancestor to have talked to him.
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Re: BROWN EYES AND RED EYES  
« Reply #246 on: May 10th, 2016, 3:37am »
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If you buy something in a shop and say you forgot something and need to return in the same shop.  You are still carrying your purchase in your hands.
 
You migh be worried that when leaving again a shop employee could think you are stealing the item.  You know you woud never ever steal anything.  But you still consider yourself in the role of a potential thief.
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Re: BROWN EYES AND RED EYES  
« Reply #247 on: May 10th, 2016, 8:16am »
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on May 10th, 2016, 3:37am, Grimbal wrote:
If you buy something in a shop and say you forgot something and need to return in the same shop.  You are still carrying your purchase in your hands.
 
You migh be worried that when leaving again a shop employee could think you are stealing the item.  You know you woud never ever steal anything.  But you still consider yourself in the role of a potential thief.

but if someone else sees entire situation (let's say person A) he doesn't think that you are a thief, moreover person B sees person A and entire situation he doesn't think person A thinks you are a thief unless he thinks person A doesn't see entire situation
 
what i fail to understand is why going every hypothetical step further we omit bit of information that is an obvious observational fact and go imagine universe that is not possible and base our conclusions on that.
 
for me it's like this in implication terms
p => q
and for p we substitute obvious false sentence
(if there are 0 REM)
now for q we substitute whatever
and it is always truth
you can say N REM die on Nth night
and i can say they don't
from my point of view it's only possible for someone to think p is possible to imagine for someone else in 1 REM and 2 REM case
 
tourist might have as well said: at least one of you have red eyes and thinks earth is flat.
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Re: BROWN EYES AND RED EYES  
« Reply #248 on: May 10th, 2016, 10:25am »
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on May 10th, 2016, 8:16am, riddler358 wrote:
what i fail to understand is why going every hypothetical step further we omit bit of information that is an obvious observational fact and go imagine universe that is not possible and base our conclusions on that.
Because it's an obvious observational fact in the world you're in, not in the hypothetical world you're considering.
 
It's an obvious observation fact that your tenth generation paternal ancestor cannot speak. Because he's dead. But all your ancestors up to 70 thousand years or so ago quite probably talked to the one before. We can imagine your 9th ancestor speaking to your tenth ancestor, even though we can't imagine either of them speaking now. Observations about the here and now don't necessarily apply to the situations we imagine.
 
When you see 2 REMs, and you try to imagine what would happen if they were the only two, then you really need to commit to the idea of the 2 REM world, look at it entirely from their viewpoint, and not from your own.
 
Suppose I can see that Alex and Bob have red eyes. If I were Bob, I would (in the 2REM case) only see that Alex had red eyes.
If I were Bob imagining to be Alex, then if my imaginary Bob was considering the 1 REM case, he can imagine Alex seeing no REMs.
Because I'm imagining that I don't have red eyes, and I'm imagining that Bob is imagining he doesn't have red eyes.  
Each step adds another red eyed monk imagining that he doesn't have red eyes. That goes on until there's none left. And then the announcement happens and some very highly hypothetical down-the-line imaginary red eyed monk's illusions get shattered.
 
If I don't know that I have red eyes, and Bob doesn't know that he has red eyes, then why would I think that Bob must somehow know that Alex can see someone with red eyes?  
If I think that Bob knows for certain that Alex sees someone with red eyes, then Bob must be thinking that Alex sees either Bob or me as a REM. So either I think that Bob does know that he's a REM, or I know that I am. Which is a contradiction.
« Last Edit: May 10th, 2016, 10:27am by towr » IP Logged

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Re: BROWN EYES AND RED EYES  
« Reply #249 on: May 10th, 2016, 12:33pm »
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on May 9th, 2016, 12:16pm, riddler358 wrote:

 
as i mentioned before there are 2 cases:
2 REM case -> both die because each of them can imagine that other one is imagining 0 REM case
3 REM case -> noone can imagine that any of other 2 are imagining 0 REM case to begin with

 
Can you ever figure out which case you're in?
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