The Hardest Logic Puzzle Ever

By John Farrier in Science & Tech on Mar 17, 2010 at 10:05 am

There’s a classic logic puzzle in which a person must take a fork in the road, one of which is safe, but the other is a deadly trap. Two men guard the fork, one of which always lies, but the other always tells the truth. The person gets to ask one question in order to determine which path is the safe one.

Mathematician Raymond Smullyan has added another dimension to this puzzle. See if you can figure it out:

There are three guardians, A, B and C. Their names are Knight, Knave and Chaos. Knight always speaks truly, Knave always lies. Chaos tossed a coin this morning to decide whether today he would behave like Knight or like Knave.

Your task is simple: ask three yes-no questions, each of a single guardian, and determine which is Knight, which is Knave, and which is Chaos. There is, alas, a complication: the guardians understand English but will answer in the local language, in which “Da” means yes and “Ja” means no. Or possibly “Ja” means yes and “Da” means no – you cannot remember.

Link via Marginal Revolution | Raymond Smullyan

UPDATE 3/18/10: Here‘s the solution.


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  1. xadrian
    Mar 17th, 2010 at 10:41 am

    I have to go back to “Will he tell me that this door leads to the castle at the center of the Labyrinth?” and then work from there.

  2. Miss Cellania
    Mar 17th, 2010 at 10:50 am

    Now I know if there are only two of them, you ask each if the other will answer truthfully. Three of them? That takes more thinking than I have time for right now.

  3. Foodie
    Mar 17th, 2010 at 10:51 am

    Hey! Aren’t these guys all Congressmen? They sure sound like they are.

  4. Him
    Mar 17th, 2010 at 10:54 am

    @Miss Cellania: How would that work? The knight would say that the knave would lie, and the knave would say that the knight would lie.

  5. Gadfly22
    Mar 17th, 2010 at 10:56 am

    Smullyan — not Sullyman.

  6. shadowfirebird
    Mar 17th, 2010 at 11:03 am

    I so nearly have the answer.

    If you ask all three guardians “How would chaos answer the question ‘is chaos telling the truth today?’ “, then you will always get Y, Y, N. N will be the Knave. And this will also tell you which of Ja/Da is yes.

    If only you had one more question to ask the Knave which of the other two was Chaos…

    Of course you might get lucky and hit an N in the first two Guardians you question; but that’s not a solution.

    Only one

  7. Chris J.
    Mar 17th, 2010 at 11:11 am

    Ask each person if the third person is Chaos. If the first two answers are the same, ask the third person if the first (or second person) is Chaos. I’m not going to list out all my handwork, but I think that system works.

  8. Chris J.
    Mar 17th, 2010 at 11:28 am

    Nevermind, my solution should be amended so that the third always answers for one or two, but I cannot resolve the condition when the first two answers are the same based on whether or not Chaos is lying.

  9. Roger Wilco
    Mar 17th, 2010 at 11:39 am

    All i know is never get involved in a land war in asia.

  10. Bryn
    Mar 17th, 2010 at 11:40 am

    Hmm, you have to ask the same person all three questions, right?

    So, the first question is “Are you a liar.” All three will answer no to that, so you know what the word for “no” is. Second question, “Will person number 2 tell me that person number 3 is a liar?” If the answer is yes then chaos is a truth teller, if the answer is no then chaos is a liar. The next question is “will person number 2 tell me that person number 3 wants me to go this way?” If chaos is a truth teller then yes means that’s the bad way, no means it’s the good way. If chaos is a liar then yes is good, no is bad.

  11. Scott Carney
    Mar 17th, 2010 at 11:49 am

    When figuring out the identity of the guardians, do you still, also need to figure out which road to take? That would require a 4th question, right?

  12. Bryn
    Mar 17th, 2010 at 11:53 am

    Jeez… i guess it would help to read the question before trying to answer it. Back to the drawing board.

  13. JJ
    Mar 17th, 2010 at 11:54 am

    With two people you ask which way to your village. The liar will point to the truthful village and the truthful one will point to his own village so they will both point the same way.

  14. Kalel
    Mar 17th, 2010 at 11:58 am

    WHAT… is your name?

    WHAT… is your quest?

    WHAT… is the air speed velocity of an unladen swallow?

  15. Bryn
    Mar 17th, 2010 at 12:11 pm

    With two people, I think the question is “which way will he tell me is safe” and the answer will be the way to danger. This question is much harder because if you go to the original post, there is a clarification that the Knight and Knave do not know whether Chaos will answer truthfully.

  16. Edward
    Mar 17th, 2010 at 12:52 pm

    At what point do logic puzzles become so convoluted and arbitrary that they pass from intriguing to nonsensical? Wherever that point exists, this one has blown past it with the speed of an unladen sparrow.

  17. John Farrier
    Mar 17th, 2010 at 1:08 pm

    Thanks, Gadfly22. Fixed now.

  18. Rich
    Mar 17th, 2010 at 1:16 pm

    The answer is 42.

  19. fsdoubleflip
    Mar 17th, 2010 at 1:23 pm

    I have part of it solved

    if you ask “does ja (or da, doesn’t matter which one) mean no?”, and the person answers with the same word, then they’re lying, so you’d be able to tell who the knight or the knave is by seeing which one is the “odd one out”

    here’s why:

    If ja meant no, and the person answered with ja, then they’d be lying, because they’d either be:

    A) saying “yes”, which, if you think about it, would contradict itself, or
    B) saying “no”, which also would contradict itself (by saying “no”, they’re confirming the question, while at the same time negating it)

    either way the person is lying. There will be two “knaves” or two “knights” when you ask the question to all three. Whoever is left out is the true knight or knave.

  20. Cluck
    Mar 17th, 2010 at 1:51 pm

    I think this is the answer:
    Ja=yes Knight Knave Chaos (truthful)
    Are you Chaos? da ja ja
    Does Ja = yes? ja da ja
    Does Da = No? ja da ja

    Ja=yes Knight Knave Chaos (lies)
    Are you Chaos? da ja da
    Does Ja = yes? ja da da
    Does Da = No? ja da da

    Ja=No Knight Knave Chaos (truthful)
    Are you Chaos? ja da da
    Does Ja = yes? ja da ja
    Does Da = No? ja da ja

    Ja=No Knight Knave Chaos (lies)
    Are you Chaos? ja da ja
    Does Ja = yes? ja da da
    Does Da = No? ja da da

    So if one answers the same for all 3 questions while the other two answer different for 1 & 2 then the one that answered all the same is chaos and Ja is yes, then you know who knight and Knave are.

    If two answer the same for all three then the one that didn’t is Chaos and Ja is no and the one that agreed with Chaos on question 2 is Knight.

    Anybody see anything wrong in the logic?

  21. Cluck
    Mar 17th, 2010 at 1:56 pm

    correction:

    If two answer the same for all three then the one that didn’t is Chaos and Ja is no and the one that said Ja with Chaos on question 2 is Knight.
    *********

    This took forever to write out the logic in words.

  22. Tylur
    Mar 17th, 2010 at 2:37 pm

    @ Cluck: No you have three questions MAX and you must ask EACH knight ONE question. You can’t ask three questions and have all three answer each one.

  23. Vonskippy
    Mar 17th, 2010 at 4:30 pm

    So questions only? No waterboarding?

  24. Avarana
    Mar 17th, 2010 at 5:16 pm

    I’d simply assume they all are Buddhas

  25. neilo
    Mar 17th, 2010 at 5:20 pm

    My brain is full. May I be excused?

  26. Captain Fatty
    Mar 17th, 2010 at 5:42 pm

    Kalel – a European swallow or an African swallow?

  27. soberdrunk
    Mar 17th, 2010 at 9:17 pm

    wow…all you have to ask is one question to all three. A question that you know the answer to (is green grass green?). You can figure out who is telling the truth from that. Then ask a fourth question to the person that was telling the truth about which path is safe. This is a logic question that states you can only ask three questions without giving the reason for that. Actually, you can ask as many questions you like.

  28. Johnny Cat
    Mar 17th, 2010 at 9:23 pm

    soberdrunk-
    What about knowing what’s yes and what’s no? And where do you get the part about asking as many questions as you like?

    I’m sure the riddle involves the tossing of the coin somehow.

  29. soberdrunk
    Mar 17th, 2010 at 9:41 pm

    Is green, green? Logic…understand logic. If someone tells you to use logic to answer the question, then anything is a possibility.

  30. Johnny Cat
    Mar 17th, 2010 at 10:47 pm

    I’ve had the rhetorical conversation before about color. Orange was the subject, and what if I perceive “orange” the way you perceive “green” and it Always boils down to empty rhetoric.

  31. Johnny Cat
    Mar 17th, 2010 at 10:52 pm

    I wish Spock was here to tell us either the answer, or to eloquently inform us as to why the puzzle itself is ILLOGICAL.

  32. stephbot
    Mar 17th, 2010 at 11:20 pm

    @brendan doherty

    LOL, yes, exactly yes. Excellent.

  33. Brent
    Mar 18th, 2010 at 2:06 am

    You stab all three and take the fourth path.

    What fourth path? The path you just came down to reach where they are.

  34. FF
    Mar 18th, 2010 at 6:32 am

    This is truly a brain twister…
    Has anyone actually found a solution?

  35. limmiboi
    Mar 18th, 2010 at 9:50 am

    Ask Guardian A, “If I asked you, “Are you Chaos?” in your current mental state, would you say “ja”?”
    If A answers “ja”, he’s Chaos. Then ask Guardian B, “If I asked you, “Are you Knight?”, would you say “ja”?”. The answer is “ja” if B is Knight, and either way the problem is now solved.

    If A answers “da”, he’s either Knight or Knave. So ask him instead, “If I asked you, “Are you Knight?”, would you say “ja”?” If the answer is “ja” he’s Knight; otherwise he is Knave.
    So, now, ask him “If I asked you, “Is guardian B Chaos”, would you say “ja”?” If he answers “ja”, B is Chaos, and C is the opposite of A. If he answers “da” then C is Chaos and B is the opposite of A.

  36. Ilan Ben Menachem
    Mar 18th, 2010 at 11:47 am

    i these this time no one tell you right path.

  37. furipu
    Mar 18th, 2010 at 7:45 pm

    If yes they nod their heads, if no they shake their heads. Local language irrelevant.

  38. NegaFalcon
    Jul 5th, 2010 at 3:03 am

    Very Easy… Ask one of them it does not matter weather he is a knight or a knave. Ask “what will you say if the other guard will point the correct path?” if the guard you asked answered “the left” go to the right.. if he answered “the right” go to left.. simple…


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