Knights and Knaves - Another Truth and Liar Problem
An island has only two types of people: Knights (who always speak the truth) and Knaves (who always lie).
I met two men who lived there and asked the taller man "Are both of you Knights?".
He replied with a "Yes" or "No", but from his answer, I could not figure out what type of person each man was.
I then asked the shorter man "Is the taller man a Knight?".
He replied with a "Yes" or "No", and after that I knew which type of person each man was.
Were the men Knights or Knaves?
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16 solutions
Yeah case 4 would be the knight saying no and the knave saying yes
Case 4 is impossible, since the knave can't tell the truth.
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he's not. He's implying that the knight is not a knight, thus he's lying
But it didn't say their responses How the hell are you supposed to tell which they are? I said that the shorter was the knave, assuming he said yes. I don't understand
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There are 2 answers. Extraneously you can tell if person 1 says yes an person 2 says no. Who they are aswell
I..I completely read the question wrong I read if the shorter was taller than the big guy. Not if the big guy was the knight. is an idiot
Yes, if the taller man was a knave, he would have said yes. BUT the problem states that from the taller man's response "I" could not figure out what each person was, which is impossible since the taller man said yes, and I would not have known if he was a knave and thus I would have assumed that both are knights, which is contradictory to what the problem states.
Well, if the taller one had replied with "At least one of us is a Knight." and if the shorter one said that the taller one isn't a knight, then it would have been clear as well.
Case 3 is impossible because the knight cannot say no because that would not be the truth.
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It seems you misunderstood the table above; the first column supposes the identity of the Taller person and their answer to the first question. The second column is for the Smaller person and their answer to the second question.
I got stuck at the point in which both are being knights by agreeing that one is taller
The TALL GUY had to reply with "NO" because he asked if "BOTH" are Knights, and IF he said YES, then it would mean the author already knows..... but since he said HE CANT DETERMINE by his first answer WHICH GUY IS WHICH...... THAT MEANS he had to say NO, AND BECAUSE HE SAID "NO" he had to make second question and ask the second guy aswell. The Correct answer is, Taller guy is Knight who said "NO" and then KNAVE said "NO" aswell. If the Taller guy was Knave and he LIED about it and answered "YES" to the question "IF BOTH of them are Knights" then the author wouldnt have to be confused, and wouldnt have to ask another question, + there is also a chance, that the second guy is a KNAVE aswell, HE COULD HAVE SAID YES ASWELL which means they would be BOTH KNAVES.... Why would the author be confused if the TALL GUY DIDNT SAY "NO"? :D he HAD to say NO, if he said YES then he wouldnt be confused and he would just assume they are both knights Why he couldnt figure out which type of person they were? if he said YES..... the True answer cant be "YES" when he says that he cant figure out their types right afterwards, my logic is that because he CANT figure out their types, BECAUSE he answered with "NO" means HE HAS TO ASK THE SECOND GUY. Btw if Author DIDNT SAY "I COULDNT FIGURE OUT WHICH TYPE THEY WERE" then obviously First man saying "YES" (knave= tall) and short guy (Knight) saying "NO" WOULD MAKE SENSE. BUT since he said that he CANT figure out which type they were, It IMPLIES that he answered with a "NO". Author shouldnt say that he was"Not able to determine their type" he simply should have said "AND THEN I asked second guy aswell" WIthout saying that he couldnt determine it, IF he just asked them BOTH without stupid comment, then this question wouldnt be a problem. (and the right answer would be Knave = tall and Knight = short, BUT SINCE HE IMPLIED HAVING PROBLEMS DETERMINING AFTER THE FIRST QUESTION, BASED ON THAT LOGIC, it IMPLIES the TALL GUY answered with a "NO"..... the reason why he said "NO" is because author says, that he was confused and couldnt determine which one is which, if you ask 2 random people if they are BOTH brothers or Cousins and they will reply with "NO" it means you have to ask again, if they answered with "YES" it means they are Both Brothers or cousins and there is no need for second question, you should have just asked them both individually without making any other comments about it.
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You are wrong here. If the first question was answer with a NO, then the asker (and us) would have deduced that the one answering that first question is a Knight and the silent one(who haven't been asked to answer anything, yet) is a knave. Please re-read the puzzle and the solution post above.
we can reach the conclusion only if shorter call the taller knave in this situation.
T: we are knaves. S: T is Knight......................result= confusion
T: we are knaves. S: T is Knave...................... result: S is surely knight. (knave cannot call him knave)
T: we are knights. S: T is Knight...................... result=confusion
T: we are knights. S: T is Knave...................... result= T is surely knave (these should be the answers to conclude)
why is the case 3 "T:we are knights . S : T is knight " results in confusion?
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Case 3: if both are knight: they are saying the truth if both are knave: they are both saying the lie if T is knight and S is knave: not true if T is knave and S is knight: not true
Therefore, in case 3, there are 2 possible truths, which will lead to confusion
Because either T and S can be knights......In case both are knaves,then also they would say that they are knights because knaves lie.So there are 2 possibilities.Hence it will result in confusion.
It can be called a situation like " a view with a jaundiced eye" in practical. Since two of the preferable answers are given as "both are knights" and "both are knaves"as a choice , the question implements a vicious circle problem, if integrating it as a solution in practical.The existing answer choosen as true, from brilliant.org can be interpreted and affirmed as "the logic of prejudgement". The question itself is not forming the vicious circle, but the answers given as a choice.
The statement is not that clear
(taller:knight, shorter:knave) ........... no ........... no
(taller:knight, shorter:knight) ........... yes ........... yes
(taller:knave, shorter:knave) ........... yes ........... yes
(taller:knave, shorter:knight) ........... yes ........... no
Basically, if the taller says Yes then it doesn't matter what the shorter response is, the person who is asking them wouldn't have enough information to know. But since the person who made the question was able to derive the truth, the taller responded No without doubt. Thus, the correct answer should be Taller=Knight, Shorter=Knave. The existing answer considered correct by brilliant.org (T=Knave, S=Knight) is incorrect.
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T says No, assume that True, then S must say No which would be False for T=Knight, S=Knave, but if S say Yes he'd be telling the truth which he could not do since he's a Knave. So a No from T has to be true in which case there is no indetermination. Also 3 out of 4 cases have T saying Yes with questioner unable to determine truth. Only the case where S says No is it clear that S=Knight and T=Knave. If S had said Yes no determination would be possible.
The fact that he needed to ask a second question means that "(taller:knight, shorter:knave) ........... no ........... no"
is not an option. IE the tall one did not say "no".
Taking the remaining 3 options, the only option that yields a unique result is the last one: (taller:knave, shorter:knight) ........... yes ........... no
Ahmad is right. If they both say yes, it's inconclusive. It is impossible for T to say no and s to say yes. But there are two (2) scenarios that will both yield definite but opposite answers and those are if T says yes and s says no, or if both say no. The answer choices do not reflect both as equal possibilities.
Heres my thought process:
First thing that narrows down the situation: "I met two men who lived there and asked the taller man if they were both Knights. He replied, but from his answer, I could not figure out what type of person each man was". The important thing is that the author doesn't know the types of the men after he hears the answer.
In the "no" scenario, the tall man cannot be a knave because if he was, he can only say yes since he is a knave himself. Therefore he is a knight, and short man is a knave because they are not both knights, and the tall man is a knight. Because now we know both types, the tall man must have answered "yes" since the author was still unsure after his reply and he would be sure if he answered no.
So now we know for sure that tall man answered Yes. If he is a knight, they are both knights. If he is a Knave, there is 1 or 2 knaves among the pair. The author says that " I asked the shorter man if the taller man was a Knight. He replied, and after that I knew which type of person each man was", so he must give an answer that leads to only one possible permutation. We don't know what short man answered. So finding the solution is a matter of testing yes/no from the short man to see which answer results in a resolution of the situation.
In the "yes" scenario, if short man is telling the truth, he is a knight and tall man is also a knight, or he is lying and they are both knaves. Therefore he must have answered no because there are 2 possible branches from the "yes" path. Now we have to figure out who is who. Tall man says they are 2 knights and short man says the tall man is a knave. Tall man must be a knave because there is contradiction and therefore one is lying (therefore one is a knave and he is lying so he's a knave). Because the short man said the tall man is a knave, he is telling the truth and therefore is a knight.
I'm not really happy with how little elimination I was able to do in this problem. Are there more efficient ways of reaching the solution? And I'm not sure what techniques from the wiki I was using to solve here.
When a person lies they are always scared and speak in light slow voice. So the man couldn't figure out what the tall man said. Then he asked the short guy who speaked the truth as he spoke with clarity and moderate voice which led the man to understand the stuff. That's how he figured out who is who.
Brilliant! oh and another thing, by calculating the odds, it is clear that to make people mistake who do not actually think logic, the asker made the short one knight, and thats how I actually did, and after that I thought the logic out of it :P
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Awesome! When you can switch something to Probability and Odds, everything becomes easier!
Sorry, but this isn't logical. You cannot assume that everyone who lies speaks the same way
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True. So how about all 3 agreeing that one of the subjects is taller?
On the basis of taller man's ans (yes/no), the possible conditions are
Yes = TT / FT / FF (T=True/Knight, F=False/Knave)
No = TF
On the basis of shorter man's ans (yes/no), the possible conditions are
Yes =TT / FF
No =FT / TF
Now from the information, after asking taller man: "He replied but I could not figure out what they were" , it is sure that taller man's ans was "yes".
& from the information, after asking shorter man, he determined what type of people they were: the shorter man's ans must be "No".
Then, from their answers the common answer is FT i.e. Taller man was lying & shorter man spoke truth.
so, taller man was knave & shorter man was knight.
Tall is Knight, Short is Knave - Tall: "No" - Short: "No"
Tall is Knave, Short is Knight - Tall: "Yes" - Short: "No"
Tall is Knight, Short is Knight - Tall: "Yes" - Short: "Yes"
Tall is Knave, Short is Knave: - Tall: "Yes" - Short: "Yes"
Notice that The last two cases have the same responses. So they can be eliminated. You can determine that Tall is Knight and Short is Knave if it is the first scenario. Since the question says you cannot, the first case is eliminated. That leaves the second case: Tall is Knave, Short is Knight.
Very easy, once everything clicked. The reason why it wasn't at first was because I didn't know how to interpret the statements:
- "...but from his answer, I could not figure out what type of person each man was." - "...and after that I knew which type of person each man was."
...because these could be taken a few different ways, but I'll focus on the way that actually solves the question.
Take all possible answers, and match them up to these two statements. There can't be any contradictions.
Case 1: The taller was a knave and the shorter was a knight.
If the taller was a knave, then it makes sense that you couldn't tell the difference between the two from his answer; this implies that he's a knave. There's no contradiction.
If the shorter was a knight, then it makes sense that you would be able to tell the difference between the two, because this would mean that the shorter would answer that the other man isn't a knight, which perfectly fits.
Case 2: They were both knaves.
If they were both knaves, then it would make sense that you couldn't tell them apart from the first answer. However, it wouldn't make sense that you could after the second answer. This is a contradiction with the narrative of the question.
Case 3: They were both knights.
If they were both knights, then it wouldn't make sense that you couldn't tell them apart from the first answer, because you would have gotten the truth. This is a contradiction with a narrative of the question.
Case 4: The taller was a knight and the shorter was a knave. If the taller was a knight, then he would have said no to the first question: They're not both knights. Which means that you would have been able to tell the difference between the two from the first question, which is a contradiction with the narrative of the question.
When you try to solve this question, what I imagine trips most people up is the narrative of the question. Usually, you try to find contradictions within the characters of the question, more than you do the characters with the narrative. But once you understand how to properly interpret the narrative, this question is actually extremely easy.
If you'd like a shortcut: the narrative says that you could tell who was who from the first question; this implies that he lied to you. You were able to tell them apart after the second question, because he told you the truth. Happy solving.
From the question it is implied that they're not both Knights nor Knaves. If after asking the first man if both are knights the man couldn't tell what was the truth, we can say that the taller man must be lying, which makes him a Knave and therefore makes the shorter a Knight by default.
Sorry, but I don't see this implication at all. You have to test each possible case, not assume.
Firstly, if the taller said that they both are knaves, then it totally sure that they are knaves, there is nothing to this.
If the Taller one said that they were both knights and I was confused, I asked the shorter, who, if said that 1) the taller was a knight(yes) , then still it would be unsure if they were knights or knaves, which is not possible by given conditions. 2) the taller is knave(no) , then it is sure that taller is knave and shorter is knight, which is the correct answer.
NOTE: If you cant get it at first shot try again by slowly reading and understanding it.
Incorrect on the first part. If the taller one said that they are both Knaves and you conclude that they are definitely Knaves from that information like you said, the taller one would have been telling the truth; Knaves cannot tell the truth. If they were lying about being Knaves, that means they are Knights. However, Knights cannot lie. There's a flaw there.
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The flaw is in a misunderstanding of the question posed. "Are you both knights?" if answered with a "no", does not mean they're both knaves. A knight would answer "no" if the shorter man was a knave. A knave would answer "yes" regardless of whether the shorter man was a knight or a knave.
It's not that the first response was confusing, it's that the first response was not a unique response, i.e., the "no" that a knight would give because the shorter man was a knave.
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Right. I believe it is implied that he simply answers the question asked with a Yes or No; he doesn't volunteer other information like "We're both knaves."
Let's designate Knight as H(onest) and Knave as L(iar).
Then we have the following table:
HH, HL, LH, LL
Y, N, Y, Y*, Both H?
Y, N, N, Y, Taller man is H?
For the first question, the taller man will answer yes in all scenarios except scenario 2 where the tall is honest and the short is a liar. Unfortunately he answered Y, and so there's no way to find out who's what. That means we can rule out scenario 2, leaving only 3 other possibilites.)
We need another question to get more information, hence the second question. The narrator found the answer after the second question. Of the three remaining scenarios, only scenario 3 is unique. Hence the solution is No. 3.
- In actuality, scenario 4 can be a unique case because assuming both are L(iars), answering yes or no to the question "Are you both H(onest)?" are equally valid. Yes - since both are L, the answer is valid. No - since he can still lie by saying only one of them is a liar However, by implying only one of them is a liar he is indirectly speaking the truth. Hence the table above.
Ahh, missed the 'first answer gave him confusion part', so I didn't get it at first. That was interesting, thanks!
There would be four possibilities as below
- Taller is Knight and Shorter is Knight
- Taller is Knave and Shorter is Knight
- Taller is Knight and Shorter is Knave
- Taller is Knave and Shorter is Knave
For all above possibilities, Taller's reply to first question would be 1. Yes 2. Yes ( Because the truth is No) 3. No 4. Yes ( Because the truth is No)
So, for the third case, in which taller replied "No", the situation gets cleared that taller is knight and shorter is knave, but the man says that from his answer I was unable to figure out the exact situation. So, the taller man had not replied as 'No'. It means he has replied Yes.
Now we come to shorter person, his reply for different possibilities would be
- Yes
- No
- No
- Yes
As we already have excluded the third possibility, so coming over to other three. Keep in mind that after the shorter man's reply, the visitor will understand their actual identity. From 1st and 4th answer, their identity can't be cleared. So, second possibility is the correct one.In reply to Naveen Dubey; see there would be four possibilities as below
- Taller is Knight and Shorter is Knight
- Taller is Knave and Shorter is Knight
- Taller is Knight and Shorter is Knave
- Taller is Knave and Shorter is Knave
For all above possibilities, Taller's reply to first question would be 1. Yes 2. Yes ( Because the truth is No) 3. No 4. Yes ( Because the truth is No)
So, for the third case, in which taller replied "No", the situation gets cleared that taller is knight and shorter is knave, but the man says that from his answer I was unable to figure out the exact situation. So, the taller man had not replied as 'No'. It means he has replied Yes.
Now we come to shorter person, his reply for different possibilities would be
- Yes
- No
- No
- Yes
As we already have excluded the third possibility, so coming over to other three. Keep in mind that after the shorter man's reply, the visitor will understand their actual identity. From 1st and 4th answer, their identity can't be cleared. So, second possibility is the correct one.
TALLER IS KNAVE AND SHORTER IS KNIGHT.
The questions are :
1) "Are both of you Knights?"
AND
2) "Is the taller man a Knight?"
Towards the first question, a Knight would have two different answers according to who his company is, but a knave would only have one answer in response to this particular question since his existence there (to answer it) has already limited the logically correct reply to be a NO, though the knave being a knave can only change it to be a deceptive YES.
By this knowledge that a NO to the first question would be a unique answer (by a Knight with a knave companion), and that the asker, "I", admitted to be unable to figure out who's who by the answer "I" got from the first man, then the only logical thing that we can deduce is that the first answer was a "YES", even though we can't really say at this point whether this yes is the truth (by a Knight) or a lie (by a knave).
For the second question, "I" was asking about the identity of the first responder to the second one, so there are four different situations depending on who's answering and who's being asked about. A "YES" could be either a Knight's reply about a taller Knight company or a knave's reply about a taller knave company, while a "NO" on the other hand, could be either a Knight's reply about a taller knave company or a knave's reply about a taller Knight company. There is nothing unique to be deduced solely by this question, but we already have another useful information that we have about this duo, and that would be the answer of the first question that helped us to eliminate the possibility of taller Knight and shorter knave pairing earlier.
We are told that "I" finally managed to find out each of their type(s) after the second response is given, then it must be because the eliminated pair that "I" got from the first question helped them to get a unique response for the second one. If we're to do the same, then we found out that after the elimination, now a "NO" reply (to the second question) could only be possible to be heard from a Knight answering about a taller knave identity.
Answer : a taller knave and a shorter Knight, who answered a "Yes" and a "No" to the questions respectively.
I solved it by looking at the four possible answers and playing the scene out in my head each time. You know there's only four possibilities, and three of them have Taller saying Yes. So the only way you'd know on the first answer what they are is if Taller said No. So you can eliminate Option 4, because he didn't know on the first answer. Down to 3 options, 2 of them have the shorter man saying Yes. That would still leave it ambiguous. Because he knew after the shorter man's answer, he had to have said No, which means there's only one possibilty: Taller said Yes and Shorter said No. Which means Taller is Knave, Shorter is Knight.
If both of them say the other is a Knight, they could both be Knaves or Knights. Similarly, if both of them say they are Knaves, clearly neither of them could be Knaves.
If the taller man says only one of them is a Knight, but the shorter man says the taller man is a Knight, it cannot be determined which of them is telling the truth.
The only consistent case that could provide the traveler with any usable information is if the taller man says they are both knights and the shorter man discredits him.
If the taller man answers No, he must be a knight because a knave's answer to this question must always be Yes. It follows then if they are not both knights and the taller man is a knight, the shorter must be a knave. At this point the narrator could have determined the answer and not asked a second question. Therefore the taller man could not have answered No.
If the taller man answers Yes, and the shorter man also says Yes, they are either both telling the truth or both lying, in which case the narrator still wouldn't have been able to tell who a knight or knave.
The taller man must have answered Yes and the shorter must have answered No. If the taller man was a knight and they were indeed both knights, the shorter man could not have lied and said the taller man was not a knight. Instead, the taller man must be a knave lying to say Yes, both men are knights, and the shorter man is a knight confirming that the taller man is a knave.
possible case & response
1 : Knight Knight (YY)
2 : Knave Knave (YY)
3 : Knight Knave (NN)
4 : Knave Knight (YN)
If 3 is the case, we can know based on first response
since we cannot determine it, so the CASE 3 is distinguished
the author know who are they based on second response, so the CASE 4 is the unique one
Lets look at all the answers for all possible situations.
Case 1:
Case 2:
Case 3:
Case 4:
In case 1 and 2, we would not have been able to determine what they are based on the responses. They could either both be Knights or Knaves. We can eliminate these cases because the author was able to determine what they were.
In case 3 and 4, we would be able to determine what they are based on the responses. The key to figuring out which one was the case is in the fact that the author was unable to determine the case after asking the first question.
If it had been case 4, then the author would have been able to determine what they were after asking only the first question since only case 4 has a difference response, No , for the first question.
But the author stated he was unable to determine after asking the first question. This implies that the response to the first question was Yes , since case 1, 2, and 3 both share that response for the first question. It was only after asking the second question that the author was able to eliminate case 1 and 2.
This means it was Case 3 .