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Another Puzzle [Oct. 18th, 2005|12:31 pm]
This is an old one that I think is rather interesting and shows a lot about how we approach logical reasoning:

The Surprise Quiz

One Friday, a professor tells his class of logic students that there will be a surprise quiz next week. It will be a "surprise" because they won't know on which day he will give it.

The students eagerly discuss this among themselves after class and some up with a startling line of reasoning:

The quiz cannot be on Friday. If it were, consider what our situation would be Thursday after class. If we haven't had the quiz by the end of Thursday, then it must be on Friday by a simple process of elimination. But this means that it will no longer be a surprise! We would have deduced when it would be beforehand. Thus, since the professor said it would be a surprise, it cannot be on Friday.

But it likewise cannot be on Thursday. We've already determined that it cannot be on Friday. So if we get to the end of class Wednesday without having the quiz, then, again by the process of elimination, the quiz must be on Thursday. (It's as if, by eliminating Friday, we've moved the end of the week closer.) Since we could determine on Wednesday night that we'd have the quiz on Thursday, though, it is no longer a surprise.

And we can continue this process through the whole week. If we haven't had the quiz Tuesday night, it must be on Wednesday; this eliminates Wednesday as a surprise. It we haven't had the quiz Monday night, it must be on Tuesday; this eliminates Tuesday as a surprise. We're left with only Monday... But it's not a surprise if that's the only possible day! So Monday is eliminated, too.

The students conclude that there can be no quiz. They write this up over the weekend and present it to the professor. Perhaps this is the "real" quiz? The professor compliments them on their work.

The professor then gives his quiz on Tuesday, as he had planned. The students are indeed surprised!

What was the flaw in the students' reasoning?

[Edit: I'm going to post my thoughts tomorrow. There are a number of ways to approach this classic problem. I'll work from the assumption that the professor did not intentionally lie. I also assume (but did not clearly state in the problem) that the professor has selected the quiz day in advance.]

[User Picture]From: kuddlepup
2005-10-18 08:11 pm (UTC)

They make the assumption that they get through each day without a test, and basing the results of the following day on that. There is no provision for them actually having the test offered in their initial variable.

Kinda right? Maybe?

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[User Picture]From: griffinwolf
2005-10-18 08:14 pm (UTC)
Well, if the class were a quantum physics class, then the professor gave them an ultimatum which by itself would be as the students would think, creating the possibility that the event of a test MUST be a surprise.

That professor is a geeeeeeeeee-nius :)
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[User Picture]From: traveller_blues
2005-10-18 08:14 pm (UTC)

Here's my guess:

The flaw is that the surprise vector is set with all the possible days (Monday through Friday) available, before any of those states are collapsed. In fact, the test vector only collapses when the test is given, because then they know there will be no test the -rest- of the week.

Therefore, by roughly the same reasoning, the test cannot be Monday, which fits the answer of Tuesday.

Admittedly, this is only true by that logic -- the test is whenever the professor gives it. Including Monday, which could be a surprise if they were expecting any day other than Monday.

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[User Picture]From: nicodemusrat
2005-10-18 09:55 pm (UTC)

Re: (real answer is the bit in italics)

Good points. :)

The last bit is the reason that I included a definition of "surprise" in the problem. (Maybe not the usual definition, as we think of it.) I'll limit what we consider to be a "surprise" to not knowing in advance which specific day he will give the quiz. (Stated alternately, if you cannot know with certainty whether there will be a quiz tomorrow, the presence of a quiz would be a surprise.)
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[User Picture]From: twopiearr
2005-10-18 10:07 pm (UTC)
the student's assumption is correct only if the test hasn't been given by Friday. If the test is not given by Wednesday, then it could be given either Thursday or Friday and still be a surprise.
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[User Picture]From: pippinbear
2005-10-18 10:54 pm (UTC)
Ahhh, the old Unexpected hanging paradox. ;>

In your version, though, the professor doesn't appear to have specified for how long the quiz will be a surprise. Maybe he only meant they won't know which day it will be on until at least Monday...

But really, it all depends on how you define surprised, and how logical you require the students to be. If the students hadn't bothered disecting the professor's statement, they could have guessed when the quiz would be, and if they'd been right, they wouldn't have been surprised! Or they could have paranoidly expected the quiz every day until it happened, and also wouldn't have been surprised when it did, in fact, happen. (Even if they were surprised on the days when they *didn't* get quizzed!)

By doing their reasoning using a definition of "surprise" which requires the students to be strictly logical, by assuming that the quiz would meet that definition of being a "surprise" and thus causing themselves to expect the quiz never to happen, they pretty much guaranteed that they would be surprised (in the everyday sense of the word) when the quiz actually happened.

In other words, it's all down to their reasoning being part of the process which decides whether they will be surprised or not! I think it makes the entire situation self-referential. If anyone else had done the reasoning, it wouldn't have affected the outcome!

Paradoxes are weird. *nod*
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[User Picture]From: rakhan
2005-10-19 03:47 am (UTC)
Surprise is an unknown, unplaned for event... by telling some one you have a surprise next week, no matter what day they know its coming, thus making it not a surprise.

Rather like someone "Planning for an emergengy" ;)
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[User Picture]From: tafyrn
2005-10-19 04:42 am (UTC)
The flaw in their reasoning is that they equate the concepts of "surprise" with "inability to predict using logic".
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[User Picture]From: xolo
2005-10-19 04:59 am (UTC)
The students' solution ignores one of the stated premises of the problem, namely, that the test will take place. They're solving a different problem than that which the professor stated.

That's the practical answer, and would probably get full marks on a test. The more interesting question is why the students ignored one of the conditions. I think the answer can be found in the way in which the professor phrases his statement. Of the two conditions stated, only one is testable. We can make an objective measurement of whether the test was given in the timeframe specified. Whether the students are surprised is a subjective response, though. There's no way to measure that objectively - we have to rely on the students' reporting whether or not they were surprised. This raises problems...

When given such a problem in such a setting, we tend to reinterpret the problem in objective terms, because that's what classical logic was designed to deal with. From the beginning, the students are defining 'being surprised because they don't know the day' as synonymous with 'being logically unable to predict the day'. This leads them into the fallacy of stating that if a day can be logically deduced as being the test day, then no-one can be surrpised when the test falls on that day, ergo that day can be eliminated. This fails to work because surprise is not objectively testable, and further depends upon variables external to the system of logic.

It is entirely possible, for instance, for a student to work the problem, and not believe his logically valid results, and therefore still be surprised when the test day arrives. Equally, there may exist a student so paranoid about being tested that no test could ever surprise him - he's surprised only when not tested.

What's happened here is that the students have become blinkered by the system of reasoning that they're using, and have failed to recognize that one condition of the problem simply can't be addressed by that system. In trying to reconcile the unverifiable condition with the verifiable one while still staying within the formal logic system, they've restated the unverifiable condition in terms that are objectively verifiable. Unfortunately, the meaning has been distorted in the process. Using the distorted condition, one CAN reconcile the two conditions by supposing that the the first one (there will be a test today) always has a value of false. That doesn't accord with reality, though.

(What Tafyrn said in one sentence while I was busy typing, basically :) )
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[User Picture]From: bogglerat
2005-10-19 07:57 am (UTC)
True the test can't be on Friday, because they'd be able to predict it on Thursday night. And true that on Wednesday night, they would therefore know it would be on Thursday.

But on Tuesday night, the chain breaks down. The test could either be Wednesday or Thursday, and neither can be eliminated.
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