Riddles & Brain Teasers
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La Canaliste
Member # 5563
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written Saturday, October 29 2005 00:54
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quote:The new information is this: If monks can see one red-eyed monk, who does not commit suicide, they believe that he does not know there are any red-eyed monks. Now they know that he does know, and he can act on that knowledge. The trigger is that knowledge has been transferred. -------------------- I am a pale shadow of the previous self. quote: Deep down, you know you should have voted for Alcritas! Posts: 387 | Registered: Tuesday, March 1 2005 08:00 |
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Electric Sheep One
Member # 3431
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written Saturday, October 29 2005 01:04
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I don't think it can quite be that, because if there are at least two red-eyed monks (REMs), then every other monk sees that all the REMs can see at least one REM. So no monk can suppose that any REM believes there are no REMs. [ Saturday, October 29, 2005 01:05: Message edited by: Student of Trinity ] -------------------- It is not enough to discover how things seem to seem. We must discover how things really seem. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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...b10010b...
Member # 869
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written Saturday, October 29 2005 01:49
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quote:But it's not sufficient for every monk to know that there's at least one red-eyed monk: every monk has to know that every monk knows there's at least one red-eyed monk (because their actions are based on the actions of other monks, and what other monks do is based on what they know). If there are two red-eyed monks, it isn't automatically the case that they know that each of them knows that there's at least one red-eyed monk, because neither red-eyed monk knows that they themselves are red-eyed, so neither of them knows that the other can see a red-eyed monk. However, once the tourist makes his statement to all the monks, it becomes the case that every monk knows that every monk knows there's at least one red-eyed monk. (An interesting consequence: if the tourist took each red-eyed monk aside in private and told them individually that there was at least one red-eyed monk, none of them would commit suicide unless there was only one red-eyed monk, as in this case no new information would be conveyed to anyone who could already see a red-eyed monk. It's necessary that the tourist make his statement in such a way that every monk knows that every other monk has heard it.) (An even more interesting consequence: suppose the tourist took each red-eyed monk aside in private and told them "There is at least one red-eyed monk. I have told every monk that there is at least one red-eyed monk, but I have not necessarily told every monk that I have told every monk that there is at least one red-eyed monk". In this case, the red-eyed monks would only commit suicide if there were 1 or 2 of them -- on the first day if there were 1, or the second day if there were 2. If there were 3 or more red-eyed monks, none would commit suicide, because each monk would simply assume that the two red-eyed monks he could see each hadn't been told that the other knew that there was at least one red-eyed monk. Consider the case of 3 red-eyed monks. Even if REMs A, B and C all know that there's at least one red-eyed monk, and know that they all know that there's at least one red-eyed monk, they can't commit suicide unless they know that they all know that they all know that there's at least one red-eyed monk. For all A knows, B might not have been told that C knew that there was at least one red-eyed monk. If B didn't know that C knew this (he actually did, but A didn't know that B knew that C knew), then B couldn't be sure that C would kill himself on the first day if he was the only red-eyed monk (remember, as far as A knows, B could think that C might be the only red-eyed monk, although B does not actually think this since he, unlike A, can see that A is red-eyed). Therefore A can't kill himself on the third day just because B and C haven't, unless he knows that B and C both know that each other knows that there's at least one red-eyed monk. Really. All of this bizarre behaviour is a consequence of the fact that the knowledge and behaviour of any given monk is dependent on that monk's knowledge that all the other monks know a particular piece of information, and what exactly this particular piece of information is becomes more complicated with the number of monks involved.) [ Saturday, October 29, 2005 02:52: Message edited by: Explode Thuryl Now ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Electric Sheep One
Member # 3431
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written Saturday, October 29 2005 03:16
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Ah, that's right, only with three or more REMs is it obvious to all Ms that all Ms know that REMs exist. With only two REMs, each REM has to allow the possibility that the other sees no REMs and thus thinks there might be none. With 3 REMS: 1 all monks know that a REM exists; 2 all monks know point 1; BUT not all monks know point 2. The 3 REMs have to allow that there might be only two REMs, each of whom would have to allow that there might be just one REM, each of whom might might have to allow that there might be no REMs. So in general, for N REMs, we have N statements that hold before the tourist speaks: 0 a REM exists n all Monks know statement n-1, where 0<n<N. After the tourist speaks statement N holds ["all monks know statement N-1"], and only then. Thus the new information learned by the monks from the tourist's statement is statement N-1. What is still curious, perhaps, is that the tourist's statement actually is just 0; yet what it conveys to the monks is N-1. Evidently the monks use a logical calculus which infers N-1 from the premise 0. Yet this can't be quite right, for they do already know 0, except in a trivial case. So why is this still confusing? That is, can we explain clearly how the monks' logic distinguishes between 0 as already known by them all, and 0 as declared by the tourist? I think Thuryl is in the right direction in pointing out that the input from which the monks deduce statement N-1 is not just the statement 0, but rather the observation that all the monks have accepted statement 0. And yet this amplified input is just adding statement 1 to statement 0. And except in simple cases, statement 1 is also already known by the monks. Our effort to understand this problem is itself coming to resemble the problem, with its peculiar induction. This is perhaps not so surprising, given that solving the REM problem requires figuring out how the monks themselves ... solve the REM problem. [ Saturday, October 29, 2005 03:38: Message edited by: Student of Trinity ] -------------------- It is not enough to discover how things seem to seem. We must discover how things really seem. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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...b10010b...
Member # 869
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written Saturday, October 29 2005 03:29
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It's the fact that all of the monks are told the information together that's the critical part. Since this happens, they can all observe that all the other monks have been told statement 0, which makes statement 1 true. And, since all of the monks have perfect and equivalent logical and observational faculties, they can know that all the other monks have observed all the monks being told statement 0, which makes statement 2 true. And so on. It's not the information that's conveyed that's important so much as the way in which it's conveyed. Now that I think about it, this all requires that the monks already have an awful lot of implicit knowledge about the capabilities of the other monks, most of which is assumed in logical puzzles but not necessarily realistic in a real-life situation. [ Saturday, October 29, 2005 03:46: Message edited by: Explode Thuryl Now ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Electric Sheep One
Member # 3431
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written Saturday, October 29 2005 03:53
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Naw, information is information. The distinction you draw is just between different channels. The metaproblem is to unpack the REM problem from its vernacular presentation, in which explicit and implicit information are treated differently, and set all information on the same footing. That is, I want to formalize the set of premises held by the monks initially, the new premise supplied by the tourist, and the rules of inference that the monks apply. Edit: we are dueling with edits. Your point about massive implicit knowledge by monks has occurred to me too, and it worries me, because I won't really be happy if the formal set of premises initially held by the monks is outrageously long - either infinite, or scaling badly with N. That clearly isn't really the way people, even smart fanatical monks, really think. Instead we apply some form of default reasoning, which generates formal premises on the fly as needed, but which hasn't itself been satisfactorily formalized, at least to my knowledge. Unless there has been a breakthrough in the past few years, default reasoning is an open philosophical problem. [ Saturday, October 29, 2005 03:58: Message edited by: Student of Trinity ] -------------------- It is not enough to discover how things seem to seem. We must discover how things really seem. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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La Canaliste
Member # 5563
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written Saturday, October 29 2005 04:07
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Fiona: I love you Charles Charles: I know Fiona: I know you know Charles I know you know I know Fiona: I know you know I know you know And so on (Goon Show sketch of hallowed memory) What the tourist did was to ensure that everybdody knew that everybody knew there was at least one REM. That was the new information. If that was different from what I said before, then I said it badly before. -------------------- I am a pale shadow of the previous self. quote: Deep down, you know you should have voted for Alcritas! Posts: 387 | Registered: Tuesday, March 1 2005 08:00 |
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...b10010b...
Member # 869
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written Saturday, October 29 2005 04:08
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Well, whatever method of reasoning they use, our first assumption must of course be that all the monks reason in the same way, and that they do so flawlessly and without error. Of course, it's also necessary that all the monks know that all the monks reason flawlessly. Question: is it necessary that all of the monks know that all of the monks know that all of the monks reason flawlessly, and so on ad infinitum? Normally this would be a silly question, but today it seems appropriate. I think the answer probably has to be yes, but I'm too fatigued to prove it tonight; I'll come back to this tomorrow. saunders: We've agreed on that much by now; at the moment we're just trying to hammer out exactly how the monks worked that out from the tourist's statement. I was happy enough with my solution about half a page ago, but I'm a biologist and SoT's a physicist. :P [ Saturday, October 29, 2005 04:11: Message edited by: Explode Thuryl Now ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Law Bringer
Member # 2984
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written Saturday, October 29 2005 04:23
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It helps to see the monks not as a single entity with information, but rather a number of entities. Together, they receive no new information - however, each is lacking a single bit of information that cannot be deduced: The color of his own eyes. Indeed, collectively, the monks know all there is to know. What prevents them from killing themselves is their vow of silence, which bars them from sharing their knowledge. When the tourist gives them the new information, he in fact tells them nothing that the monks did not know, but he tells them something the monks did not know the others knew, and that they cannot tell each other. If a single monk were to spray-paint the walls, it would have the same effect. However, in the context, that would be breaking the vow of silence, since it is a communication between the monks. -------------------- The Encyclopaedia Ermariana <-- Now a Wiki! "Polaris leers down from the black vault, winking hideously like an insane watching eye which strives to convey some strange message, yet recalls nothing save that it once had a message to convey." --- HP Lovecraft. "I single Aran out due to his nasty temperament, and his superior intellect." --- SupaNik Posts: 8752 | Registered: Wednesday, May 14 2003 07:00 |
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Shock Trooper
Member # 5969
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written Saturday, October 29 2005 04:46
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quote:I think so. If there is one red-eyed monk, and he hears that there is at least one red-eyed monk, he has to be able to figure out that he's the only one and kill himself. (If that doesn't happen, and all the other monks are logical enough to figure out that that means there must be at least two, and none of them see another one, then the whole monastery except for the stupid guy would be dead.) If there are two red-eyed monks, each has to be smart enough to see that when the other one doesn't kill himself, he must be the other one. I suppose that would require each of them to know that the other one was capable of that logic, otherwise they might think each other to be like the guy who can't figure it out above. If there are three, each of them has to know that the others can figure out that, if two days have passed and nobody is dead, he must have red eyes. But if Monk X sees two other red-eyed monks, but doesn't know that Monk Y knows that Monk Z can figure it out logically, then he might think that Monk Y didn't kill himself because Monk Y thought that Monk Z didn't kill himself because he couldn't deduce the colour of his eyes, and there isn't necessarily a third one. -------------------- A C, an E flat, and a G walk into the Tower of the Magi. Ambrin runs up to them and says, "Hey, look! It's the Triad!" Kelner snorts and says "Pretty minor Triad if you ask me." Posts: 242 | Registered: Thursday, June 16 2005 07:00 |
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Electric Sheep One
Member # 3431
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written Saturday, October 29 2005 05:27
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quote:My pendantry is over the precision that it is the (N-1)th iteration of knows that knows that ... which is new. Just 'everybody knew that everybody knew' is only enough for N=3. I guess the sort of answer that would make me happy would be an algorithm for the monks' reasoning. It could for instance use a function Belief(m,n) which gives the belief of monk #m about the color of the eyes of monk #n. Perhaps it would need a whole stack of Belief(m,n,p,q,...) for the belief of monk q about the belief of monk p about .... The question is, what is the simplest simulation of the REM problem, which would take as inputs only the number of REMs and a single bit representing whether or not the tourist has spoken, which would generate the sequence of monk populations on subsequent days. Actually, that is not the question, for the simplest such algorithm just spits out what we know to be the right answer, which is a simple formula, without in any way simulating the monks' reasoning. And this observation raises a whole new bunch of questions. Murg. -------------------- It is not enough to discover how things seem to seem. We must discover how things really seem. Posts: 3335 | Registered: Thursday, September 4 2003 07:00 |
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Lifecrafter
Member # 6403
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written Saturday, October 29 2005 11:13
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the REMs are getting a little boring, try a 7x7 square, numbers 1-49, all lines adding up to 175 [ Saturday, October 29, 2005 11:23: Message edited by: Infernal666hate ] -------------------- I've remained focused on what i believe in, which is melodic metal. It's not just a passing phase, it's a way of life! -Jon Schaffer And when I laugh it's tears I hide And when I cry it's joy inside A foul disease has stained the land The bitter harvest of a dying bloom And when I cry it's joy inside A wicked smile for all the tears I hide It hurts to hold all the pain I feel The bitter harvest of a dying bloom Posts: 883 | Registered: Wednesday, October 19 2005 07:00 |
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Post Navel Trauma ^_^
Member # 67
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written Sunday, October 30 2005 03:14
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The information the tourist provides is: (1) There is at least one red-eyed monk (2) All the monks know that (1) is true (3) All the monks know that (2) is true (4) All the monks know that (3) is true etc If there are n monks with red eyes, then (n) (and anything beyond that, although it isn't relevant) is new information to them. The non-red-eyed monks already knew (n), but they didn't know (n+1) or above. -------------------- Barcoorah: I even did it to a big dorset ram. desperance.net - Don't follow this link Posts: 1798 | Registered: Thursday, October 4 2001 07:00 |
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Shake Before Using
Member # 75
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written Sunday, October 30 2005 07:40
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Shamelessly done with Google in about 5 seconds. Posts: 3234 | Registered: Thursday, October 4 2001 07:00 |
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Lifecrafter
Member # 6403
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written Sunday, October 30 2005 12:45
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you need a machine to do it in 5 seconds, 30 39 48 01 10 19 28 38 47 07 09 18 27 29 46 06 08 17 26 35 37 05 14 16 25 34 36 45 13 15 24 33 42 44 04 21 23 32 41 43 03 12 22 31 40 49 02 11 20 very simple formula. -------------------- I've remained focused on what i believe in, which is melodic metal. It's not just a passing phase, it's a way of life! -Jon Schaffer And when I laugh it's tears I hide And when I cry it's joy inside A foul disease has stained the land The bitter harvest of a dying bloom And when I cry it's joy inside A wicked smile for all the tears I hide It hurts to hold all the pain I feel The bitter harvest of a dying bloom Posts: 883 | Registered: Wednesday, October 19 2005 07:00 |
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Shake Before Using
Member # 75
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written Monday, October 31 2005 06:57
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Well, what I did was to Google for one that already existed, rather than bother generating one myself. :P Posts: 3234 | Registered: Thursday, October 4 2001 07:00 |
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E Equals MC What!!!!
Member # 5491
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written Monday, October 31 2005 12:29
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I just read all that Monk stuff, and am so glad that it all got figured out before I did. Otherwise, I would have been pulling my hair out all day. -------------------- Sex is easier than love. Posts: 1861 | Registered: Friday, February 11 2005 08:00 |
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Shock Trooper
Member # 5969
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written Monday, October 31 2005 12:55
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This is one of those riddles where there's a specific situation and you ask yes or no questions to figure out what exactly is going on... ^_^ (I actually made this one up, so it's probably either really obvious or just really bad. ^^; ) A man is sitting in a dark pit. There are winds blowing all around him, and in front of him is the imposing figure of a man with a stick. After an hour or two, it gets light again, and the man gets out of the pit and goes home. What was he doing? -------------------- A C, an E flat, and a G walk into the Tower of the Magi. Ambrin runs up to them and says, "Hey, look! It's the Triad!" Kelner snorts and says "Pretty minor Triad if you ask me." Posts: 242 | Registered: Thursday, June 16 2005 07:00 |
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E Equals MC What!!!!
Member # 5491
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written Monday, October 31 2005 12:58
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Playing in an orchestra. -------------------- Sex is easier than love. Posts: 1861 | Registered: Friday, February 11 2005 08:00 |
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Shock Trooper
Member # 5969
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written Monday, October 31 2005 13:01
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quote:>< Gah. Told you so. -------------------- A C, an E flat, and a G walk into the Tower of the Magi. Ambrin runs up to them and says, "Hey, look! It's the Triad!" Kelner snorts and says "Pretty minor Triad if you ask me." Posts: 242 | Registered: Thursday, June 16 2005 07:00 |
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E Equals MC What!!!!
Member # 5491
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written Monday, October 31 2005 13:26
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Your sig brands you as a music geek, so from there it was pretty easy. -------------------- Sex is easier than love. Posts: 1861 | Registered: Friday, February 11 2005 08:00 |
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? Man, ? Amazing
Member # 5755
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written Monday, October 31 2005 14:10
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There is a row of five houses, each having a different colour. In these houses live five people of various nationalities. Each of them nurtures a different beast, likes different drinks and smokes different cigarettes.Briton lives in red house.Swedish has a dog.Dane drinks tea.Green house is just left to the white house.Proprietor of green house drinks coffee.The one who smokes Pall Mall, has a bird.Proprietor of yellow house smokes Dunhill.Fellow from the central house drinks milk.Norwegian lives in the first house.Blend smoker lives next to cat lover.Horse breeder lives next to Dunhill smoker.Beer drinker smokes Blue Master cigarettes.German smokes Prince.Norwegian lives next to blue house.Blend smoker has a neighbour, who drinks water.Who has fish at home? *this message sponsored by ghougle* Posts: 4114 | Registered: Monday, April 25 2005 07:00 |
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Shock Trooper
Member # 5969
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written Monday, October 31 2005 14:18
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quote:^_^ The suffix "-gry", though it can be found in the words "angry" and "hungry", is not otherwise a common one. There are only three words in the English language. What is the third one? It's something we use every day. [ Monday, October 31, 2005 14:22: Message edited by: Explore the Surface Now ] -------------------- A C, an E flat, and a G walk into the Tower of the Magi. Ambrin runs up to them and says, "Hey, look! It's the Triad!" Kelner snorts and says "Pretty minor Triad if you ask me." Posts: 242 | Registered: Thursday, June 16 2005 07:00 |
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...b10010b...
Member # 869
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written Monday, October 31 2005 14:32
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"Language". Side note: there are actually about a dozen English words that end in "-gry", although most are archaic, obscure, or variant spellings. [ Monday, October 31, 2005 14:33: Message edited by: Explode Thuryl Now ] -------------------- My BoE Page Bandwagons are fun! Roots Hunted! Posts: 9973 | Registered: Saturday, March 30 2002 08:00 |
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Shock Trooper
Member # 5969
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written Monday, October 31 2005 14:39
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Hence the sentence "It's something you use every day". *I* don't use an aggry every day... maybe a gry, though. >< [ Monday, October 31, 2005 14:46: Message edited by: Explore the Surface Now ] -------------------- A C, an E flat, and a G walk into the Tower of the Magi. Ambrin runs up to them and says, "Hey, look! It's the Triad!" Kelner snorts and says "Pretty minor Triad if you ask me." Posts: 242 | Registered: Thursday, June 16 2005 07:00 |