A reasoning puzzle– an answer

by Greg Mayer

Reader Chris G. sent a video of Steve Pinker explaining the very same problem (another reader also mentioned that he had seen a Pinker talk about it). Here it is.

I’ll post my own discussion later today.

h/t Chris G.

19 Comments

  1. Garnetstar
    Posted April 16, 2018 at 12:56 pm | Permalink

    So, once again, we see that no text is not subject to interpretation.

    Pinker doesn’t ask one question, he asks two, and the “reader” must interpret from the context and intonation, which he means. He first asks “What is the smallest number of cards that you have to turn over from this array to verify whether the rule does or does not hold of this deck of cards?” The answer (that he means) to that question, is, two.

    Later he changes the question: “Which are the cards that you must turn over to test whether the rule does or does not hold?” Now he wants you to name the cards, not say *how many*.

    His meaning, though, is clear, in that, as a speaker, he can now lead us to answer the second question, not the first.

    But, the second question can be variously interpreted as well: what does he mean by “test”? Does he mean “which card(s) would just test the statement, not necessarily definitively prove or disprove it” or does he mean “which card(s)must be turned to definitely prove or disprove it?”

    That is not at all clear, and while many people might guess the interpretation that he wants (the second one, as his answer shows), *many* others might read the word “test” as not the equivalent of “prove definitively”, as very often the verb “test” is used to only mean “probe something”, not “prove it”. They would therefore say “only D” or “only 7” are “the cards that you must turn over to test (not prove, in their interpretation) whether the rule does or does not hold?”

    In fact, which interpretation he means is not at all clear until he explains his answer. I was split between the two meanings, and I would have asked him “What do you mean by ‘test’? In your first question you said “verify”, but now you’ve abandoned that question and asked a different one. Do you mean us to take “verify” fron the first question as the equivalent of “test” in your second?

    There is not text that is not subject to interpretation by the reader or listener. There is no question that can be asked that wholly tests only your intelligence or your reasoning ability, and does not also test your ability to guess correctly what the author or speaker means.

    • Garnetstar
      Posted April 16, 2018 at 1:07 pm | Permalink

      And, of course, if this phrasing of questions was written down and read, not delivered by a speaker, everyone reading it would be completely at sea. The speaker’s intonation and body language and sequencing are giving some information as to his meaning, whereas, in writing, this phrasing would be incomprehensble to a reader.

      • Craw
        Posted April 16, 2018 at 9:09 pm | Permalink

        Piffle.
        And in his discussion Pinker is illustrating a useful thinking technique: replacing one question with another one to help understand the problem.

        • Garnetstar
          Posted April 17, 2018 at 11:19 am | Permalink

          No, Pinker doesn’t re-phrase the question for clarity, which is what you mean. He asks a completely different question, which has a different answer.

          If this was a written problem, with those two questions printed exactly as he asked them, and “2” was a possible answer, how would one know what he was asking? They are two wholly different questions.

          And yes, “test” does not always mean “prove”. “We tested our hypothesis” does not necessarily replace “We proved our hypothesis.”

          All texts require interpretation, and, based on your knowledge about who’s asking a question and so what the speaker or writer is probably meaning, people will do better (or worse) at answering them.

          You should see how many interpretations of “See Dick run” can have!

  2. Curt Nelson
    Posted April 16, 2018 at 1:14 pm | Permalink

    I don’t understand why 7 is needed — because if there was a D on the other side it would disprove D then 3.

    Can’t the same thing be said for H, that a D on the other side would disprove D then 3?

    I agree with Garnestar, that the problem was awkwardly set up.

    It seems to me that to prove the statement one would have to turn over the D, the H, and the 7.

    • Curt Nelson
      Posted April 16, 2018 at 1:49 pm | Permalink

      I meant D, F, and 7.

      • Curt Nelson
        Posted April 16, 2018 at 1:53 pm | Permalink

        Oh, a number on one side and letter on the other (so no need to look at F). Never mind.

        • Garnetstar
          Posted April 16, 2018 at 4:16 pm | Permalink

          Well, I meant, Pinker’s explanation wasn’t too bad, better than the one originally asked (see my post on that question).

          I meant, there is no text that doesn’t require interpretation. “See Dick run” must be interpreted acccording to your best guess as to what the writer meant.

          So, these questions are never wholly “tests of deductive reasoning”. They should be called “tests of deductive reasoning along with the inevitable tests of how good you are at guessing which interpretation I mean.”

    • Posted April 16, 2018 at 1:49 pm | Permalink

      It’s got to be a number on one side and a letter on the other so H can’t have a D on the other side.

      Unless H is lower case (Planck’s constant) in which case it’s a number.

      • Curt Nelson
        Posted April 16, 2018 at 1:53 pm | Permalink

        Thanks!

      • Jessy Smith
        Posted April 16, 2018 at 2:10 pm | Permalink

        When Pinker states: “Every card has a number on one side and a letter on the other. Here is a rule…”

        Are not both rules?

        A cynic might suggest that if one rule is unsure, how do we know the other rule isn’t as well?
        It’s what initially tripped me up with the puzzle. I guess I’m just not that trusting.

    • Michael Fisher
      Posted April 16, 2018 at 1:55 pm | Permalink

      ** Conditions: 4 cards are displayed & we are told to assume that each card has a numeral on one side & a letter on the other side

      **The cards: D F 3 7
      ** We are asked which cards we need to turn over to prove this statement is True or False:

      “letter D cards always have a numeral 3 on the other side”

      [note that we’re NOT asked to prove the obverse: “numeral 3 cards always have a letter D on the other side”]

      D: We need to turn this card
      If it displays a non-3 numeral the statement is FALSE

      F: No need to turn this card
      Because it would only prove the obverse

      3: No need to turn this card
      Because it would only prove the obverse

      7: We need to turn this card
      If it displays a D letter the statement is FALSE

    • S.K.Graham
      Posted April 16, 2018 at 3:06 pm | Permalink

      It is already given that each card has a number on one side and a letter on the other.

      If *that* rule is called into question, all cards must be examined.

      But given that rule, you must turn over only and exactly the D and 7 to confirm that the D–>3 rule is not broken by any of the present cards. The other two cards are incapable of breaking the rule.

  3. Mark Sturtevant
    Posted April 16, 2018 at 2:50 pm | Permalink

    One can pretty much see from this what the answer will be for our current puzzle. I anticipate learning that I am wrong, but also that my solution was not the most common wrong answer. I am one of the rare types of idiots! 🤔

  4. nicky
    Posted April 16, 2018 at 3:16 pm | Permalink

    Again I miss in the comments how ‘difficult’ the card problem is, and how ‘easy’ the exact same problem is in the context of ‘cheating’.
    I think the problem has not much to do with which cards to turn, but why the card problem is kinda ‘hard’, while the same ‘beer’ problem is so very easy/

  5. Truevo
    Posted April 16, 2018 at 9:09 pm | Permalink

    From a logical point of view we should always remember that “if A then B” is logically equivalent to “if not B then not A”. That explains all.

  6. Steve Gerrard
    Posted April 17, 2018 at 12:15 am | Permalink

    I find the comments with alternative interpretations of the problem fascinating. I see the task as it was meant right away; it is intriguing to me that others see it as something quite different.

    As a software developer, I quickly think of how I would do it if there were 400 cards, and I needed a program to check them all for compliance with the rule. The steps:

    If it is an E, turn it over to check for a 5.
    Else If it is a letter, it’s okay (it can have any number).
    Else If it a 5, it’s okay (it can have any letter).
    Else it is a number, turn it over to check for an E.

  7. Posted April 17, 2018 at 12:34 pm | Permalink

    Doing it in “imaginary mode” is different from the original task, as I realized when I wrote an elementary logic exam with a Wason task as a question. (I never used the exam as the course was cancelled, but…)


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