The Hardest Logic Puzzle Ever (And How To Solve It)

It's that strange time year, the lull between Christmas and New Year, when you're not really celebrating but not really working either. So, how about you wrap your brain around the world's hardest logic puzzle to keep yourself amused? Y'know, just for fun.

New Scientist has a lovely feature (which is available to read if you sign up for a free account) in its Christmas issue about the world's most difficult logic problem. If you're wondering what could possibly be so tough, check it out:

"Three gods A, B and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language in which the words for 'yes' and 'no' are 'da' and 'ja', in some order. You do not know which word means which."

It was dreamt up — and solved — by US logician George Boolos shortly before his death in 1996. What makes it so difficult is it's incredible amount of problems, all squeezed into one puzzle: language barriers, untruthfulness, and randomness, too. Philosophers claims that cracking the puzzle reveals the true nature of logic itself to those willing to toil with the problem — but if you can't be bothered, you can find a (relatively) simple solution here. [New Scientist]

Image: a r b o under Creative Commons license


Comments

    Hmm, that solution isn't really helping me. I'm stuck at the embedded question lemma on p. 2.

    Not sure why but I have this in my recent browser history already: http://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever

    Wasnt this a scene from the Labyrinth?

    Ahhh Bowie in tights & make up.... Thank god for Jennifer Connolly !

      Thank God indeed. Jennifer Connolly > * (well... just about)

    Why does guideline 1 contradict the question?:
    (B1)It could be that some god gets asked more than one question (and hence that some god is not asked any question at all)." This is in direct opposition with the question, where "each question must be put to exactly one god." One question per god, no more, no less.

      Anyway, the double-question idea from the solution is a good one. With help from Wiki, my walkthrough below.

      Q1: Ask god B, "If I asked you 'Is A Random?', would you say ja?"

      Answer "ja"
      - could mean "yes" or "no"
      -- IF "ja"=yes and "da"=no THEN
      --- IF True, his actual answer to 'Is A Random?' would have to be "ja" for him to answer "ja", meaning A is Random
      --- IF False, his actual answer to 'Is A Random?' would have to be "da" for him to answer "ja", meaning the truth is "ja" - i.e. A is Random.
      -- ELSE IF "ja"=no and "da"=yes THEN
      --- IF True, his actual answer to 'Is A Random?' would have to be "da" for him to answer "ja", meaning A is Random
      --- IF False, his actual answer to 'Is A Random?' would have to be "ja" for him to answer "ja", meaning the truth is "da" - i.e. A is Random.

      Answer "da"
      - could mean "yes" or "no"
      -- IF "ja"=yes and "da"=no THEN
      --- IF True, his actual answer to 'Is A Random?' would have to be "da" for him to answer "da", meaning A is not Random
      --- IF False, his actual answer to 'Is A Random?' would have to be "ja" for him to answer "da", meaning the truth is "da" - i.e. A is not Random.
      -- ELSE IF "ja"=no and "da"=yes THEN
      --- IF True, his actual answer to 'Is A Random?' would have to be "ja" for him to answer "da", meaning A is not Random
      --- IF False, his actual answer to 'Is A Random?' would have to be "ja" for him to answer "da", meaning the truth is "da" - i.e. A is not Random.

      THEREFORE "ja" = A is Random (and B is not Random), "da" = A is not Random (and B is possibly Random).

      Q2:If I asked you "are you False?" in your current mental state, would you say "ja"?
      If the answer to Q1 was "ja" ask C, then you will have B by elimination. If "da" ask A.

      True:
      - If "ja"= yes, then "da" would be his actual answer to "are you False?" and so being True "da" would be his answer to the question.
      - If "ja"= no, then "ja" would be his actual answer to "are you False?" and so being True "da" would be his answer to the question.
      False:
      - If "ja"= yes, then "da" would be his actual answer to "are you False?" and so being False "ja" would be his answer to the question.
      - If "ja"= no, then "ja" would be his actual answer to "are you False?" and so being False "ja" would be his answer to the question.

      Therefore if the god answers "da" it is True. If "ja" it is False.

      Q3: if Q1 was "ja", you're done as you have Random (A) and the identity of C, and B by elimination. If Q1 was "da", ask C "If I asked you 'Is B Random?', would you say ja?" (i.e. a modified Q1). As above, an answer of "ja" = B is Random, "da" = B is not Random (and C is Random).

      I think the statement is not saying you can only ask each god 1 question, but that you can only ask a question of single god, eg you can't say 'Whichever god is Random, say ja", and expect a response. You have to ask a specific god a specific question, but you can ask the same god all 3 questions.

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