One of my favorites, re-interpreted

It’s Friday afternoon and fourth period is almost over.

“Okay, just a reminder — there will be a surprise quiz sometime next week on the last three chapters!” the teacher brays as the bell rings and the students begin to shuffle out.

In the hallway, Alice looks to Bob and says, “Dude, what a crock. Mrs. Trunchbull sucks. I hate surprise quizzes!”

“Me too! You can waste the whole week studying. I’d rather just cram, take the text, and expunge.”

But then a thought occurs to Alice. “Wait a minute — she said there’s going to be a surprise quiz sometime next week, right?”

“Right…” Bob slams his locker shut, and they walk down the hall towards their fifth period class.

“But we know the test can’t be next Friday, or else there won’t be a surprise!” Alice exclaims.

“What do you mean?” Bob asks.

“Well, say she’s secretly planning on giving the test on Friday. But we know that the test is sometime next week only. So by Thursday afternoon, when she hasn’t given the test yet, we’ll know that the test has to be on Friday — no surprise!”

“Dang, I see.” Bob’s eyes open wider. “Hold on — it’s crazier than that. It can’t be Friday, right? But then it also can’t be Thursday either! Since when Wednesday afternoon rolls around, and she hasn’t given the test yet, we’ll know it can’t be Friday — as you just explained — so it must be Thursday. But if we know it has to be Thursday, that would eliminate the surprise too!”

“Wow, this is awesome. And by the same logic, it can’t be Wednesday either…”

“… or Tuesday …”

“… or Monday!”

“Holy smokes… it seems like she can’t possibly surprise us,” Bob says with a grin.

Alice frowns. “And yet I still have no idea when the quiz actually will be.” The bell rings for the start of fifth period, and the pair make their way into the classroom. “We’ll have to figure out what this means after class.”

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21 Responses to One of my favorites, re-interpreted

  1. f18225 says:

    not sure if this is related to the actual answer but i find it to be a matter of semantics? it’s not a surprise that there will be a quiz at some point next yet, but the actual date of the quiz is unknown. the students are looking at the situation like it is a surprise that there will be a quiz, but that’s not a surprise because the teacher has already announced that there will be a quiz. what the teacher really said is “there will be a quiz next week. it could be any day.”

    • aj says:

      I think the students don’t think of of the surprise as there being a quiz, but instead as the actual date the quiz will be given.

      In other words, a quiz is a surprise quiz if the students did not know whether it would be given on the day it was given. (I think this is the standard definition.)

      The students are claiming that this surprise element won’t be satisfied, since (for instance), if the quiz hasn’t been given by Thursday, they know it will be given on Friday.

  2. gdogg says:

    Hrmm…

    Do you have a simple explanation for this? I have never seen one that is satisfactory, and it turns out that a buttload of papers have been written about this.

  3. cychan says:

    I think there’s a difference between the condition that the date of the quiz is unknown at the present time and unknown at all points in time up to the moment the quiz is presented. If we take the latter interpretation, Alice and Bob’s logic seems correct. But if we take the relaxed definition of “surprise”, Friday becomes fine.

    • We have a winner!

    • aj says:

      Right. But does their logic actually mean anything, in terms of useful information? For instance, if you need to study the night before the quiz to do well on it, what night(s) should Alice and Bob study for the quiz to ensure they do well, given their analysis?

        • aj says:

          Exactly. So in what sense does their logic “seem correct”, if the surprise is still maintained (in the strict sense of “the test might be tomorrow so I have to study tonight”) until Thursday afternoon? e.g. Using the latter interpretation, at what point in time before the quiz is presented do they know the date of the quiz?

          (And if they don’t, then what is the flaw in their logic?)

          • cychan says:

            Okay, how about this: the students only proved that if they use the assumption that the statement, “There will be a surprise quiz” (in the strict sense of surprise) is true in their decision making process, then that implies that the same statement is false. All that shows is that they can’t use the assumption to make a prediction, because doing so would make the assumption false. They did not prove that “if they don’t make a prediction then the statement would be false.” This is what happens: they can’t make the assumption, so they can’t make a prediction, and the statement becomes true when they are surprised.

          • aj says:

            How is this different from proof by contradiction?

            e.g. Theorem: There will be no surprise quiz.
            Proof: By contradiction. Assume the opposite, that there will be a surprise quiz. Then (…), implying that there will not be a surprise quiz. Contradiction.

          • cychan says:

            Because I’m not proving that there will be no surprise quiz. I’m saying that there will be a surprise quiz, AND the student’s logic doesn’t have a flaw because their logic only proved that they can’t make a prediction, not that the statement itself is intrinsically false.

          • aj says:

            Sounds good to me. You are a now a master of surprise tests!

          • cychan says:

            As good as it may sound, I’m sure there’s something wrong with my explanation… it seems there are way too many papers published about this paradox for it to have a simple resolution.

          • aj says:

            This is most likely true. But the reason I like this brainteaser is because there are so many seemingly plausible explanations (and so many seemingly plausible ways to refute them). You get points for plausibility :)

          • induction on a faulty base case.

            we know that if they get to friday without a quiz, there will be a quiz on friday, “surprise” or not. they assume otherwise, making the whole inductive argument bunk.

            alternatively, observe that the “day on which the test happens” is a random process over a simple filtration, do some basic calculations, observe that they have no information at all – i.e., there’s a uniform distribution over the remaining days in the week.

          • aj says:

            Definitely clear that any other analysis shows that they don’t know very much. But could you clarify the first thing — what assumption are they making for their base case, exactly, and why is it bogus?
            (And what is their base case, also?)

            (To be clear: I’m not asking all these questions as someone who knows the answer and is trying to lead everyone in the right direction. On the contrary, I’ve seen a couple of explanations for this puzzle and it’d be cool if we came up with a better one.)

          • But could you clarify the first thing — what assumption are they making for their base case, exactly, and why is it bogus?

            Their base case is “the test can’t be on friday, because it wouldn’t be a surprise”.  This is false, based on an incorrect semantic reading of “surprise” – in possible world semantics, the correct reading has the claim about “surprise” scoped to the information available at the time of the utterance.

            i’ll post more when i’m not drunk.

          • aj says:

            No, post more when you ARE drunk. Then you can start writing “Alice and Boob” and “surprise testes” with impunity.

  4. ratatosksv says:

    I didn’t bother reading through everyone else’s explanations, but I thought I’d throw mine out.

    The conclusion that the game cannot be on Friday rests on the following premises:
    1) The quiz was not on Monday.
    2) The quiz was not on Tuesday.
    3) The quiz was not on Wednesday.
    4) The quiz was not on Thursday.

    These premises cannot be known until Thursday after class, but will always be known Thursday after class, so long as the quiz has not yet been given.

    The logical regression that the students use to conclude that the quiz cannot be on Thursday relies on the conclusion that the quiz will not be on Friday. But prior to after class on Thursday, premise (4) cannot be known. Therefore it cannot be known that the quiz cannot be on Friday.

    And without being able to conclude that the test cannot be on Friday, there is no way to conclude that the test cannot be on Thursday. And so the logical chain is broken.

    I think the “paradox” is better used to discuss aspects of knowledge than of logic itself.

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