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I.M._Smarter_Enyu
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New puzzle -- Surprise Quiz
« on: Jul 26th, 2002, 12:56pm » |
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The professor for class Logic 315 says on Friday: "We're going to have a surprise quiz next week, but I'm not telling you what day... if you can figure out what day it will be on, I'll cancel the quiz."
The students get together and decide that the quiz can't be on Friday, as if the quiz doesn't happen by Thursday, it'll be obvious the quiz is on Friday. Similarly, the quiz can't be on Thursday, because we know it won't be on Friday, and if the quiz doesn't happen by Wednesday, it'll be obvious it's on Thursday (because it can't be on Friday). Same thing for Wednesday, Tuesday and Monday. So it can't be on ANY day, so there's no quiz next week!"
They tell the professor, who smiles and says, "Well, nice to see you're thinking about it."
On Tuesday, the professor gives the quiz, totally unexpected!
What's the flaw in the students' thinking?
This is similar to the Red Eye/Brown Eye Monks problem.
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Alex
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Re: New puzzle -- Surprise Quiz
« Reply #1 on: Jul 26th, 2002, 2:25pm » |
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The students mistake probability with certainty. Each logic step they make is based solely on the step before it, without "the big picture." Their first conclusion that it cannot happen on Friday is a relatively sound assumption, but still an assumption and not a certainty. Therefore, to assume that a quiz presented on Thursday would be "obvious" if no quiz has occured by Wednesday is false, since it could still occur on Friday.
I'm afraid I'm missing the link between this and the red/brown-eye monk problem, perhaps a hint? =)
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S. Owen
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Re: New puzzle -- Surprise Quiz
« Reply #2 on: Jul 26th, 2002, 6:20pm » |
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We have a chain of reasoning like this:
If quiz hasn't happened by Thursday --> Quiz won't happen on Friday
If quiz hasn't happened by Wednesday and won't happen Friday --> Quiz won't happen on Thursday
putting them together:
If quiz hasn't happened by Thursday --> Quiz won't happen on Thursday
You don't have anything like a hard conclusion that it won't happen on Friday, or Thursday... only empty, circular statements like above. That is, the conclusion that it won't happen on Thursday is based on the assumption that it hasn't happened by Thursday!
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william wu
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Re: New puzzle -- Surprise Quiz
« Reply #3 on: Jul 28th, 2002, 4:29pm » |
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on Jul 26th, 2002, 12:56pm, I.M._Smarter_Enyu wrote:The professor for class Logic 315 says on Friday: "We're going to have a surprise quiz next week, but I'm not telling you what day... if you can figure out what day it will be on, I'll cancel the quiz."
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Cool riddle! Nice initial shock value too; I was pretty dumbfounded for a while. I have added this riddle to the archive and I would like to credit you by your real name; however, I don't have this information. I tried e-mail, but I guess that was like sending e-mail to a recycle bin (spamhole account). Currently you are listed as [I.M._Smarter_Enyu]. If you would like to send me your name, feel free to e-mail me at wwu@ocf.berkeley.edu.
Thanks!
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Joe Marshall
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Re: New puzzle -- Surprise Quiz
« Reply #4 on: Jul 29th, 2002, 4:40am » |
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Hi,
I think Scrowen has mischaracterised the students' argument. It is correct that the students start with
"If quiz hasn't happened by Thursday --> Quiz won't happen on Friday".
However, they can then conclude unconditionally that the quiz will not happen on Friday, as follows:
"Suppose the quiz happens on Friday. Then, by the end of Thursday, we would know that the quiz would be on Friday (by our previous statement), and would expect it. This braeaks the rules, so the quiz can't hapen on Friday".
This unconditional conlusion ("Quiz can't happen on Friday") eliminates the circularity which Scrowen worries about.
I think that the real error in the students' reasoning is that they have not fully considered their own mental states. They should start their argument not with
"(Hasn't happened by end Thursday) --> (Won't happen on Friday)",
but rather with:
"((Hasn't happened by end Thursday) && (We believe it will happen)) --> (Won't happen on Friday)"
If the students stop believing that the quiz will happen, then the proof breaks down!
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S. Owen
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Re: New puzzle -- Surprise Quiz
« Reply #5 on: Jul 29th, 2002, 6:05am » |
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OK, another way to say it, like yours, is this:
Remember that they use similar reasoning to conclude that it wont' happen Thursday ("Suppose it happens on Thursday. Then by the end of Wednesday..."). They also rely on the conclusion that it can't happen Friday here.
But, if by supposition, the quiz happens on Thursday, then their first argument about Friday is no longer valid ("Then by the end of Thursday [when the quiz hadn't happened] we'd know it is on Friday")! Without that part too they cannot conclude it won't happen on Thursday.
The two arguments are entwined in a non-obvious way, but the circularity it creates is the real source of this seeming paradox.
on Jul 29th, 2002, 4:40am, Joe Marshall wrote:
I think Scrowen has mischaracterised the students' argument. It is correct that the students start with
"If quiz hasn't happened by Thursday --> Quiz won't happen on Friday".
However, they can then conclude unconditionally that the quiz will not happen on Friday, as follows:
"Suppose the quiz happens on Friday. Then, by the end of Thursday, we would know that the quiz would be on Friday (by our previous statement), and would expect it. This braeaks the rules, so the quiz can't hapen on Friday".
This unconditional conlusion ("Quiz can't happen on Friday") eliminates the circularity which Scrowen worries about.
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Joe Marshall
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Re: New puzzle -- Surprise Quiz
« Reply #6 on: Jul 29th, 2002, 7:27am » |
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Sorry Srowen (spelt your name wrong last time - apologies!), but I still think you're missing something. You seem to be suggesting that in the case where the quiz happens on Thursday, the students' conclusion that it won't happen on Friday is no longer valid. But if the quiz happens on Thursday, then obviously it doesn't happen on Friday!
Similarly, if it doesn't happen on Thursday, then the original argument is valid. Either way, the quiz still can't be held on Friday (unless, as I note in my first post, the students stop believing that the quiz will happen at all - in my opinion, the real source of the 'paradox').
Any thoughts?
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anshil
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Re: New puzzle -- Surprise Quiz
« Reply #7 on: Jul 29th, 2002, 11:46am » |
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Hi william wu, first I think it would be fair to move this riddle to "very extremly super hard" category, and additionally add the "I don't know the solution myself flag". or at least a satisfactory solution. Additionally add a new "a puzzle of humanity" or "no human knows yet the solution" flag.
I've done a bit of google search on this puzzle, and well let's you about of historical view. (I do believe that history is something important) I first thought it maybe better not to tell about this, but I think it's fairer to do so. This is a very classical paradox, it first appeared around 1943. There are even whole books written just for this puzzle, and yet most seem to agree that non of the solutions provided is actually satisfactory.
Okay any company asking this in an interview is just crazy 
Well I did study physics for some time, and well they do the same practice in higher classes. On some tests or exercises the professor gives a task that is not yet solveable with today knowledge, hoping one day a smart student will solve it anyway, and of course the university gets credit for it.
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So he said to god who dissolved in a cloud of logic.
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anshil
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Re: New puzzle -- Surprise Quiz
« Reply #8 on: Jul 29th, 2002, 11:56am » |
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Anyways thanks for the riddle.
Let me introduce a simplified puzzle, which I think would have a similar solution:
Professor CrazyDazy is known to be a little crazy and confused. One day he tells his students: "Tomorrow will be a surprise examination". One studend raises his hand, and calmy explained to the professor that his proposition can't be true, since he just told them that it would be tomorrow and no suprise anymore. He agrees, and of course the studends did not prepare but went on a party. However the next day Professor CrazyDazy makes an examiniation, and imagine how surprised the studends were.
Now what happened, was the statement of the professor actually yet true? Where is the logic conflict?
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S. Owen
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Re: New puzzle -- Surprise Quiz
« Reply #9 on: Jul 29th, 2002, 12:07pm » |
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Yeah, I think what I posted before is flawed.
I have seen the classic version of this problem before, but it seems to be stated a little differently, where the professor would say something like: "there will be an unexpected quiz one day next week" and "you will not be able to logically deduce what day it takes place."
That's a bit different and more well-defined than what we have here, and I thought it might afford a solution (maybe along the lines of what the students believe, yes). But no luck from me...
I still don't see how this is like the red-eyed monks problem.
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anshil
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Re: New puzzle -- Surprise Quiz
« Reply #10 on: Jul 29th, 2002, 12:13pm » |
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Maybe I can throw in some additonal hints.
Say the professor would have said "tomorrow or the day after tomorrow is a surprise test". Well on the next day it would still be valid surprise. Since the studendts don't know if it is this day, or the next. Since if the test isn't that day it will be on the next, but it's still a surprise on that day. However on the next day the test won't be a suprise anymore, therefore render the professors argument invalid.
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william wu
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Re: New puzzle -- Surprise Quiz
« Reply #11 on: Jul 29th, 2002, 2:10pm » |
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on Jul 29th, 2002, 11:46am, anshil wrote:| Hi william wu, first I think it would be fair to move this riddle to "very extremly super hard" category, and additionally add the "I don't know the solution myself flag". or at least a satisfactory solution. Additionally add a new "a puzzle of humanity" or "no human knows yet the solution" flag. |
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Oy ... OK. Yea, actually I lied when I said I was dumbfounded for a while. I'm still very confused and disturbed.
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bartleby
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Re: New puzzle -- Surprise Quiz
« Reply #12 on: Jul 30th, 2002, 10:00am » |
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OK I just wasted like an hour here at work Googling up research on this problem.
I found a lot of sites that tried to give rigorous logical or mathematical proofs of answers to this paradox, but none of them really "felt" intuitively right, or were plain-spoken enough for me to accept.
So, what the heck, here's my crack at it:
I think the fundamental flaw is with the statement of this paradox, in the definition of the word "surprise." Specifically, the teacher's use of the word "surprise" isn't the same as the sudents', as applied over time.
I define surprise as "You do not have 100% certainty of ADVANCE KNOWLEDGE of the quiz."
Now, the teacher is using the word "surprise" to mean "You do not know with certainty RIGHT NOW what day the quiz will be next week." The students are applying the word "surprise" 5 different times, for different days of the week.
This isn't sounding as strong as I had wanted... Let me restate the problem a little bit... let's say instead of having the teacher say "If you can figure out what day it will be on, I'll cancel the quiz," he says "If you tell me what day it will be next week, and you can PROVE to me it will be on that day, I'll give you $100 right now." Now the parameters have changed slightly, because the students have to prove RIGHT THEN what day the quiz will be on.
The certainty that the quiz WON'T be on Friday isn't a CERTAINTY until the end of class on Thursday. It isn't a certainty on Sunday night!
I think I can see why this problem is like the Red Eyes/Blue Eyes problem, too, and why the proposed "answers" to the Red Eyes/Blue Eyes problem are WRONG!
Imagine the week is only 1 day long... this translates to only 1 blue-eyed person on the island... Follow this logic forward.
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anshil
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Re: New puzzle -- Surprise Quiz
« Reply #13 on: Jul 30th, 2002, 10:15am » |
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The orignial problem is formulated simpler, not opening this loop hole. The professor just tells the students there will be a surprise examination next week. [suprise in the meaning it will be a surprise the moment he deals out the papers.]. One stutend goes to him and explains the reasoning above. There just can't be a surprise examination on Friday, it wouldn't be a surprise then. So Friday is ruled, so on Thursday also can't be a surprise examination, since it wouldn't be a surprise, and so on.
Or just let's "modify" view the riddle in this way, it's far more interesting then.
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bartleby
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Re: New puzzle -- Surprise Quiz
« Reply #14 on: Jul 30th, 2002, 11:20am » |
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How does taking away the "reward" (giving the students a reward if they figure out the day it's on) change anything? Pointing out "The quiz can't be a surprise" implies "So therefore there can't be a surprise quiz" so explicitly stating "If you can negate the surprise, there is no quiz" doesn't really change anything, it's still the same paradox.
Actually, the "original" original puzzle was an announcement by the Swedish Civil Engineers in like 1954, something along the lines of "There will be a surprise evacuation drill one day next week." Then a mathematician/logicial thought about that statement and the paradox it creates.
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anshil
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Re: New puzzle -- Surprise Quiz
« Reply #15 on: Jul 30th, 2002, 1:22pm » |
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You're actually right.
It may be arrogant for me to say, but I will join the lines with heißenberg, who said anybody claiming that he understood quantumn mechanics has no idea of it. I say anybody providing an easy answert to this puzzle did not understood the problem.
I think we all concentrate too much on the students, maybe we should focus the professor more and his forecast that there will be a surprise test in the next n days. (Actually it doesn't matter how big n is), the statement itself is actually itself invalid, which becomes most obvious when considering n to be 1. (( Do you think I could say: on anyday I expect a test _could_ come, it cannot be a surprise?)) However the statement of the professor becomes valid as soon I (as student) consider it to be invalid. However it is invalid itself when I (as student) condiser it be valid. So my mind state changes reality?
Maybe his statement is something similar like another classic, like "This sentence is false". Now it can neither be true or false (in the sense of the opposite is true). Thats where many of the other "paradoxes" drain from, assuming false means the opposite is true. The sentence is just invalid, or like the sign on the bus station "The last bus will not drive". This sign is also just invalid. or the classic "The barber in a certain village is a man who shaves all and only those men in the village who do not shave themselves." Also this sentence is just invalid.
However in the case of the professor it's different as the sentence can neither be valid nor invalid.
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Chronos
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Re: New puzzle -- Surprise Quiz
« Reply #16 on: Aug 6th, 2002, 1:12am » |
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Everyone seems to be assuming that a surprise must remain a surprise forever. This is obviously not so: The day after a quiz, the students certainly know what day the quiz was. At some point, the cat must be let out of the bag. So, suppose that the test is on Friday. By the end of Thursday, the students will have figured that fact out, but up until then, they don't know. It's still a valid surprise, even if the surprise is sprung before they actually take the quiz.
This is similar to what bartleby was saying: It's a surprise because they don't know at the beginning of the week.
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Brian
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Re: New puzzle -- Surprise Quiz
« Reply #17 on: Aug 6th, 2002, 9:44am » |
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If you want a longer treatment of this puzzle, Smullyan covers it in _The Unexpected Hanging_.
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Brian
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Re: New puzzle -- Surprise Quiz
« Reply #18 on: Aug 6th, 2002, 10:10am » |
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on Jul 29th, 2002, 4:40am, Joe Marshall wrote:Hi,
However, they can then conclude unconditionally that the quiz will not happen on Friday, as follows:
"Suppose the quiz happens on Friday. Then, by the end of Thursday, we would know that the quiz would be on Friday (by our previous statement), and would expect it. This braeaks the rules, so the quiz can't hapen on Friday".
This unconditional conlusion ("Quiz can't happen on Friday") eliminates the circularity which Scrowen worries about.
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Even that first step, that seems unassailable, is wrong. Suppose the students are cautious, and decide that maybe the induction step is shaky, but *surely* the initial step is correct: if we haven't had a test by the end of Thursday, then any test will have to happen on Friday, and then it can't be a surprise. Therefore we can't have a surprise quiz on Friday.
The prof gives the quiz on Friday, and of course the students are surprised, since they concluded it couldn't happen then.
The paradox, I think, is in the attempted modelling of expectation (not in the statistical sense of the word, but in the plain English sense). "I'm going to surprise you" means "I'm going to do something you don't expect". In guessing what I'm going to do, your expectations are both axioms and conclusions, and it's that feedback that causes the paradox. It's not so very different from the old Cretan paradox, "Everything I say is a lie".
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George Wright
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Re: New puzzle -- Surprise Quiz
« Reply #20 on: Aug 9th, 2002, 4:58pm » |
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Wow this is really an amazing coincidence. I just found out about this site, a few days ago, and noticing the surprise quizz question wasn't asked, I made a note to myself to put it up. just I was just now logging on to suggest it when I found someone beat me to it. Here are my thoughts on it.
I think the problem is that the idea of surprise is inherantly self referential, and so can lead to inconsistencies. Much in the same way that the statement
"this statement is false" is a paradox.
If we try to generate the axioms of the logical
system on which the problem is based we come
up with the following
1) There exists 7 days
2) There is a quiz on one of the 7 days
3) If the quiz is on day n, then there is no way
to prove the quiz is on day n based on axioms 1)
through 3), and based on the information
given in days 1...n-1.
Axiom 3 really saying if a proof of X exists than X is false.
Which is a statement that is clearly likely to cause problems.
Somehow this also seems reminicent of Godel's
incompleteness theorem. I wish I remembered that
course I took in mathematical logic.
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zarathustra
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Re: New puzzle -- Surprise Quiz
« Reply #21 on: Sep 2nd, 2002, 8:22pm » |
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I've thought about it a lot and I think I have the simple reason why the proof fails. The major problem is that an important piece of the logic is glossed over in every step. Heres the first step simplified: If it's Thursday and the test hasn't happened yet then it must be on Friday. Clearly it must be Thursday for this peice of logic to hold true. The next step is bad logic: If it's Wednesday and the test hasn't happened yet then it must be on Thursday since it can't be on Friday based on step one. We can see that the first step cant be applied to step two becase the first step is only true on Thursday and the second step asserts that it's Wednesday. Every step continues this bad logic which causes the paradoxical conclusion. Anyway that's how I think about it.
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