So there are five beautiful ladies whose faces are veiled. You can distinguish them well as they have written their (correct) names Alice, Betty, Cindy, Diana and Emily on their T-shirts.

  • You know that two of them have brown eyes, and always tell the truth.
  • You know that three of them have blue eyes, and always lie.

Alice, Betty and Cindy are brought in front of you. You may ask each of them them a single question. Based on the three answers, you must guess the eye colors of all five ladies correctly. How do you manage to succeed?

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    I also think this puzzle needs more limitations. I could just ask each one "Are you female?" for example, and I would easily know who is telling the truth and who's not. – user14478 Nov 28 '15 at 16:22
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    @LuxxMiner yes but you only have 3 girls in front of you and you need to know the eye color of the remaining two.And you can't just say that the two of them have blue and brown eyes you must say exactly which girl has which eyes. – fmm24 Nov 28 '15 at 16:48
  • You should clarify that the remaining two ladies also have to be identified without asking them any questions. I surmise that those two ladies are in sight (meaning they could be distinguished from each other by clothes or something) and have to be identified. To clarify the meaning even further, you should say that all the ladies will then be assembled in a line (you can distinguish in between them, lets say they are wearing differently colored clothing) and then have to be identified as blue-eyed or brown-eyed – AvZ Nov 28 '15 at 18:00
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    There are 10 possibilities for the distribution of the 5 ladies. There are only 8 was ways that three questions can be answered if the questions have only yes or no answers. So it cannot be done with just yes/no answers. So if you ask "what binary number represents the five of you, where 1 is truth teller and 0 is liar" would the liar invert the answer or mix it up at random? Could the liar tell the truth in some occasions? – Dr Xorile Nov 29 '15 at 2:00
  • @DrXorile: I would assume that if one asks a question with N possible answers, a liar would determine honestly which answer is correct, and then select one of the N-1 other answers in whatever fashion would be most vexing. – supercat Dec 3 '15 at 21:24

You ask Alice, Betty and Cindy

"What are the colour of Diana and Emily's eyes, in that order?"

Reasoning:

If ABC are all liars, they will answer "blue, blue".

If two of ABC are truth-tellers, then the truth-tellers will say "blue, blue", while the liar will say "brown, brown".

If one of ABC are truth-tellers, the truth-teller will say either "brown, blue" or "blue, brown", while both the liars will say the opposite.

By counting the answers you get you should be able to deduce what colour the eyes of the girls are.

  • This is a pretty interesting aproach. And it should be a correct one since the question didn't specify wether your 3 questions can ask for multiple answers or should only be answered with true or false. Also welcome to puzzling stackexchange. – The Dark Truth Dec 1 '15 at 13:12
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    I think that a liar could be free to give any incorrect response. Thus, if ABC are liars the correct answer would be "brown, brown", so "brown, blue" would still be a lie. Stipulating otherwise essentially means you're asking each girl two questions. – user3294068 Dec 1 '15 at 13:33
  • TDT: Thank you :) user3294068: Unfortunately, you make a valid point. – kevinpieter Dec 1 '15 at 13:49
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    @kevinpieter: I would suggest that for something to be considered "one question", there should be a discrete number of possible answers, of which exactly one is correct; a liar should then be able to choose any incorrect answer. Note that this puzzle is solvable under such constraints, but your solution could fail. T=Brown F=Blue, if Alice and Betty say FF and TF, then possible layouts include FFFTT, FFTFT, TFTFF, and FTFTF. The first two questions have revealed a lot of information, but not enough for any question one might ask Cindy to be sufficient. Cindy will always be able... – supercat Dec 2 '15 at 16:45
  • ...either to answer FT or to answer FF; answering FT would eliminate TFTFF but leave the other three possibilities; answering FF would eliminate FFTFT but leave the other three. Thus, Cindy's answer won't reveal very much. – supercat Dec 2 '15 at 16:49

A nasty thing about liars is that when asked complex questions, they may answer in whatever fashion, other than 100% truthfulness, would be most vexing. That is a substantial complication which other answers fail to take into account, but the problem is still solvable as stated.

Assuming T identifies the brown-eyed girls and F the blue-eyed ones, ask Alice...

"Is the sequence one of TFTFF FFFTT TFFTF TFFFT FTFTF FTFFT?"

After that, ask Betty...

"Is the sequence one of FTTFF FFFTT TFFTF TFFFT FTFTF FTFFT?"

If both answer "yes", then ask Cindy...

"Who else has brown eyes?" You know Cindy has brown eyes, and she will truthfully identify the other brown-eyed person.

Otherwise, ask Cindy:

"Does Emily have Brown eyes?"

Cindy will be lying, but her answer along with those of Alice and Betty will reveal everything you need to know:

If either Alice or Betty says "yes" and the other "no", the one that said "yes" has brown eyes and the other blue; Diana will be opposite Emily, and Cindy will say (falsely) which of D/E have blue eyes. If neither says yes, their eyes match, and Diana's eyes will match Emily's. Cindy's answer will indicate (falsely) whether AB or DE is the blue-eyed pair.

  • This works for all 10 possibilities. Very nice. – f'' Dec 2 '15 at 0:44
  • Great answer! Follow-up question: Is there also a non-adaptive solution (where your later questions must not depend on the answers to earlier questions)? – Gamow Dec 2 '15 at 10:29
  • @Gamow: Adaptive questioning is required. One needs to get maximum information from anyone who might be lying, and the only way to do that is to ask a yes/no question. It may be possible to compose other puzzles of this form where extracting less than maximal information from a liar would suffice, and thus a non-adaptive approach could work, but I don't believe that's the case with this puzzle as given. – supercat Dec 2 '15 at 14:18
  • @Gamow: I added a puzzle with slightly easier constraints, for which a non-adaptive solution is possible (see link in side-bar). – supercat Dec 2 '15 at 19:26
  • Good answer. It looks to me like you can get away with the two-element sets {TFTFF,FFFTT} and {FTTFF,FFFTT}. You might be able to ask them that in plain english :-) – deep thought Oct 8 at 22:52

Only one question needed

Assuming we are not restricted to T/F questions, we could ask just one question, "What would one of the other 4 with an opposite eye colour to yours say are the people in this group with blue eyes?" A liar would invert the truth, and a truth-teller would truthfully report an inversion. In either case, the named people would have brown eyes and the remainder, blue.

Using @kevinpieter's idea of counting, the (single) question can be simplified to just "Who are the people in this group with blue eyes?" - if three people are named, the named people have blue eyes and the other two, brown; if two people are named, the named people have brown eyes and the other three, blue.

  • I think you meant the other three, blue – Bishop Dec 1 '15 at 22:07
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    Could not a liar respond with any arbitrary combination of eye colors which didn't precisely match the truth? – supercat Dec 1 '15 at 23:22
  • @Bishop Thanks - corrected. – Lawrence Dec 2 '15 at 0:04
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    @supercat You're right. However, it is customary in truth/lie problems for those designated as liars to give as their answers the complement of what truth tellers would say. Those who give inconsistent answers are often labelled knaves or inconsistent, but not liars. – Lawrence Dec 2 '15 at 0:13
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    @Lawrence: Puzzles of this genre often require questions with yes/no answers, If one wishes to go beyond that, I would suggest that the best formulation is to require that each question have a fixed set of possible answers, of which exactly one is correct. The liar would then be free to select from among the incorrect answers. Asking a truth-teller a multiple-choice question would give more information than a yes/no, but asking a liar a multiple-choice question would give less information. – supercat Dec 2 '15 at 14:36

There are 10 possible ways that TTFFF can be ordered, so this is provably impossible if we ask questions with only yes/no answers (as 3 questions can only distinguish $2^3 = 8$ scenarios). However, nowhere in these rules has it been stated that we may only ask such questions. Thus, I will go ahead and blatantly ask a ridiculously complicated compound question.

Question 1, to Girl 1: Assuming the color brown was defined as "true" and the color blue was defined as "false", XORing your answer for each girl with the truth value of the statement "Girl 1 Always Lies", what truth value would the eyes of each of the five girls here individually be defined as?

For every girl where she says "True", they have brown eyes. For every girl where she says "False", they have blue eyes. Since we asked her to respond individually, she must lie or tell the truth for each individual sub-answer, and in every case her lie/truth is canceled out by the XORing process, leaving only the truth. We're done, in a single question!

Alternatively, if you like slightly simpler questions:

Question 1, to Girl 1: Is it true that "Girl 2 has brown eyes" XOR " you always lie"?

Question 2a, to Girl 2: What is the pairing of names and eye colors for the five girls here?

Question 2b, to Girl 2: What is not the pairing of names and eye colors for the five girls here?

If you got a "Yes" for question 1, ask 2a. If you got a "No", ask 2b.

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    I think it's fairly optimistic to assume that these beautiful ladies have a working knowledge of logic operators – DrunkWolf Nov 29 '15 at 8:26
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    This does not work. (1) If she must lie or tell the truth for each individual sub-answer, then you are actually asking five individual sub-questions. But the rules only allow you to ask a single question. (2) Hence the five informations together form a single answer. A truth-teller will give five correct informations, whereas a liar will provide five informations, of which at least one is false. – Gamow Nov 29 '15 at 15:54
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    Oh, I am absolutely asking five separate sub questions, carefully disguised as a single overall question. My claim is that the puzzle is otherwise unsolvable. I can change the ways in which I am breaking the standard set of question-asking rules, but the fact remains that 10 > 2^3, so we're going to be doing something strange. As far as Logical operators, one can expand that statement into plain English, but I'm assuming that we on the puzzling board are familiar with them, and can do that expansion ourselves. – Zerris Nov 29 '15 at 16:26
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    You should be able to determine what #3 is with simple questions to #1 and #2, and then just ask #3 for the distribution? – DrunkWolf Nov 30 '15 at 9:56
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    @Zerris: The problem is solvable even if liars are allowed to tell arbitrary partial fibs. – supercat Dec 1 '15 at 23:09

Going off what kevinpieter wrote:

... and this doesn't work since you can't tell Diana and Emily apart.

Ask the three girls "Do Diana and Emily both have brown eyes?"

If all three answer the same thing, we know all three of them are the same type (and thus liars, which means they must have all said "No").

If two say the same thing and the third person says another thing, we know one of two things:
1) Two truth tellers and one liar were asked (in this case we know the truth tellers said "No" and the liar said "Yes")
2) Two liars and one truth teller were asked (in this case we know the liars said "Yes" and the truth teller said "No")

  • If I have a better answer, should I edit this one, or post a new one separately? – Daphne B Dec 2 '15 at 14:42

I have an answer that attempts to get around the whole "liars can give any incorrect response" problem, while also being a bit easier to ask than supercat's answer (though maybe not as good)

First, ask Alice...

if Diana has blue eyes

Then, ask Betty...

if Emily has blue eyes. Each of these questions will determine if the colors of the two girls match. Because of this, we can use the numbers of each group to determine Cindy's eye color.

If it is brown, ask Cindy...

Who else has brown eyes?

If it is blue, ask Cindy...

Of the others here, which two would never speak the truth to each other? (The negative in the question is my best attempt at ensuring any answer that isn't strictly opposite the correct scenario can't be given.)

Cindy will then point out all the remaining brown eyed girls.

  • I guess the limitation in puzzle is you can ask Single question to all three of them , so this wouldnt be appropriate? – Karun Kamal Oct 9 at 7:14
  • You can ask the first question to all 3, but it only works for some of the possible scenarios. I think it may be impossible to come up with one yes/no question that will give you enough information for all three. I was just trying to come up with a simpler version to SuperCat's solution. – Yessoan Oct 9 at 23:30
  • I don't think the last question works... because she can pick a blue-eyed girl and brown-eyed girl, which is also a lie. Though maybe I didn't understand the question ... maybe Cindy won't either ;-) – deep thought Oct 11 at 3:15
  • The way it was phrased was to invert the typical scenario where there is one correct answer and multiple incorrect answers. Two people will only speak the truth to each other if of them have brown eyes. If the last question was asked to a brown eyed girl, there would be multiple correct answers, any combination of two blue eyes or one of each would be correct. Because of this, there is only one incorrect answer, which would be the two brown eyed girls. And because we know Cindy has to give an incorrect answer, we know who she will pick. – Yessoan Oct 11 at 22:51
  • Got it. I got confused thinking they "never speak the truth to each other" if both of them have blue eyes instead of either. – deep thought Oct 12 at 23:21

Ask Alice one question:

Treating blue-eyed girls (liars) as all having the value 0 and any brown-eyed girls (truthtellers) as respectively having the values: you (Alice) 1, Betty 2, Cindy 4, Diana 8, and Emily 16, what is the sum of all girls' values?

Assuming that the lying girls can only comprehensively lie (and cannot mix truth and falsehood together so as to deceive more subtly), the number given will, when converted to binary, show a bit pattern containing either three zeroes and two ones, or three ones and two zeroes. Given that we know there are three liars, the liars are represented by the digit, be it 0 or 1, with three occurrences, and the truthtellers are represented by the digit with two occurrences.

Examples of Alice's possible answers:

 17, 10001: the two 1s represent truthtellers, so Alice and Emily have brown eyes.
 19, 10011: the two 0s represent truthtellers, so Betty and Cindy have brown eyes.
  9, 01001: impossible because Alice cannot falsely say there are 3 liars.
 22, 10110: impossible because Alice cannot truthfully say there are 3 truthtellers.

You just need two simple questions. Also, this question needs less male gaze :p

Ask Alice,

If I were to ask you whether Betty had brown eyes, would you say yes? No matter what eye color Alice has, she will answer yes if and only if Betty has brown eyes.

Now, ask Betty,

What are [not] the eye colors of all five of you? If Betty has brown eyes, you just ask her what everybody's eye colors are, and you're done. If Betty has blue eyes, you ask her what are not everybody's eye colors, so that only the correct answer will be a lie. Either way, Betty will report the eye colors of all five.

  • 1
    (edited because I misread at first) Couldn't Betty, if she was a liar, just say "Blue." since it's not the color of everyone's eyes? – Daphne B Dec 1 '15 at 21:48

@supercat says "A nasty thing about liars is that when asked complex questions, they may answer in whatever fashion, other than 100% truthfulness, would be most vexing. That is a substantial complication which other answers fail to take into account, but the problem is still solvable as stated."

I agree! I suggest asking each girl one of the following two simple english questions:

"Who is the other girl with brown eyes?" or "What is your name?"

Note that

If you ask a girl with brown eyes "who is the other girl with brown eyes?", she will just give the right answer. What happens if you ask a blue-eyed girl this question? She will do something other than naming another brown-eyed girl.

Let's start:

"Alice, who is the other girl with brown eyes?" If Alice points to Cindy, ask "Betty, who is the other girl with brown eyes?"
If Alice points to Betty, ask "Cindy, who is the other girl with brown eyes?"
If Alice points to Diana or Emily, you can ask either Betty or Cindy.

If two girls are now pointing to the same girl, or one girl is pointing to a second girl who is pointing to a third girl, then all three have blue eyes (because there are only two brown-eyed girls).
Otherwise the two girls are pointing at two other girls. One of the girls pointing, and the girl she is pointing to, have brown eyes. The other pair have blue eyes. The fifth girl, not pointing or pointed to, must have blue eyes.

You have one girl left to ask a question (Cindy or Betty, depending).
If she is the fifth girl, ask her "who is the other girl with brown eyes?" and she points to a blue-eyed girl, the other pair have brown eyes.
If someone is pointing at her, ask "what is your name?". If she says her name, she and the girl pointing at her have brown eyes, if she says something else, the other pair have brown eyes.

Now it is also possible that they would answer

"I don't know", "nobody else has brown eyes", "I am a fish", or whatever. These are also lies. Whoever does this has blue eyes. So if Alice says "I don't know", then ask "Betty, who is the other girl with brown eyes?"
If Betty points to Alice, then she also has blue eyes.
Otherwise either Betty and the girl she pointed to have brown eyes, or, Betty and the girl she pointed to have blue eyes.
Then you can ask, "Cindy, what is your name?" to tell which is which.
But if Betty pointed to Alice, you can ask, "Cindy, who is the other girl with brown eyes?"
If Cindy points to Alice or Betty, she has blue eyes, and so Diana and Emily have brown eyes. Otherwise, she and the girl she points to have brown eyes.
Of course, Betty could also say "I don't know," which also tells us she has blue eyes, do the same thing as if she had pointed to Alice.
Same if Cindy says "I don't know."


Actually, I was not 100% truthful. If you look closely you'll see I have a different interpretation than @supercat.

So let me anticipate some objections. As I say, "who is the other girl with brown eyes" poses no issue for the brown-eyed girls. I imagine one could object that: if Alice has blue eyes, then, Alice is still lying if she names Betty, even if Betty does have brown eyes ... because Betty is merely "an" other girl. (Boo, hiss.) Okay, but we can pedantically reword the question to get around this. Like "Alice: out of Betty, Cindy, Diana, and Emily, name one girl with brown eyes". This is still something Alice would answer straightforwardly if she had brown eyes.
I think the actual two interpretation differences are,
(1) Is it a lie to give a correct answer when there are actually multiple correct answers?
(2) Is it a lie to say something which is obviously incorrect?
As demonstrated, it is possible to solve the puzzle even if liars might say "I am a fish" to any question.

Well, even if you disagree with my interpretation, hopefully you appreciate the solution anyway :-)

  • By your logic, asking "Give me a pair of ladies among those here--possibly including yourself--containing at least one lady with the same eye color as you" would force someone to name the two brown-eyed ladies. One of the brown-eyed ladies would indicate a set containing herself and the other brown-eyed lady, while a blue-eyed lady would have to answer with the only pair not containing any blue-eyed ladies (i.e. the pair of brown-eyed ladies). – supercat Oct 9 at 18:25
  • If the lady responds with something other than a pair of ladies, one could repeat the process until either someone responds by naming a pair of ladies, or all three ladies have failed to do so. It seems rather unsatisfying, however, to say that as soon as one receives a responsive answer to a question one will know everything, and otherwise one will simply keep asking until someone is responsive. – supercat Oct 9 at 18:31
  • @supercat - Why can't the brown eyed lady name a pair containing a brown-eyed lady and a blue-eyed lady? That would be a pair containing "at least one lady with the same eye colour as" the one questioned. But I might be having some trouble parsing the question :-) – deep thought Oct 10 at 23:59
  • As for unsatisfying, I don't know. If there really is a question along these lines that works, why not. – deep thought Oct 11 at 0:03

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