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Topic: Pop Quiz Riddle (Read 78482 times) |
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Speaker
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Re: Pop Quiz Riddle
« Reply #150 on: Sep 2nd, 2004, 12:28am » |
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The question asks, what was the problem with the students reasoning. Well, it seems straight forward, if you ignore all the previous threads (which is easy, because I only read some of them, sorry  ).
The teacher asked the kids to guess a day.
They guessed no day (although they just said no test).
This guess was wrong, so the teacher gave the test.
The students believed (incorrectly) that if the test were not a surprise, it would be cancelled. This was the student's mistake.
What the teacher said was that, if the students could guess the day, the test would be cancelled.
The students were just too clever for their own good.
Well, I just finished reading some more of the posts. It seems to me that Blingo, on page two had the answer people wanted to get using logic. But, he only posted once, two years ago.
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| « Last Edit: Sep 2nd, 2004, 12:45am by Speaker » |
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Speaker
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Re: anyone know the answer to the pop quiz riddle?
« Reply #151 on: Sep 2nd, 2004, 12:48am » |
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Here is the post I mentioned above. It seems that the students assume conditions that are true for the last day of the week to be true for the other days of the week. Which seems to be the essence of bling0's post.
on Aug 10th, 2002, 1:14am, bling0 wrote:Ok, this was a question in a CS textbook, but I don't remember which one. I do remember, however, that this was under the induction section.
Note that the student's logic is as follows:
Base case:
IF (a) it is Thursday AND there has been no test, THEN we know that the test cannot be on Friday.
Inductive step:
Given that we know that the test cannot be on Friday, then we can apply induction to the same thing for Thursday.
Conclusion:
there can be no test
Ryan said it earlier I think. The student's *logic* is completely faulty because the student's base case does not hold. The logic relies on the fact that IF it is Thursday AND there has been no test. What if it is Thursday AND there has been a test? The logic doesn't hold. |
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rmsgrey
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Re: Pop Quiz Riddle
« Reply #152 on: Sep 3rd, 2004, 3:09am » |
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on Sep 1st, 2004, 10:52pm, honkyboy wrote:| I never said that the students were convinced that there was not going to be a test on Friday. |
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But the problem statement does - the students have to be convinced there cannot be a test on the Friday before they can use their reasoning on the Thursday - if the students aren't convinced there won't be a test on the Friday, then on the Wednesday evening, they cannot know the test must be on the Thursday, so they can't then go on to eliminate Thursday...
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Patashu
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Re: Pop Quiz Riddle
« Reply #153 on: Sep 3rd, 2004, 4:12am » |
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One problem with all of this: There can be a test on Friday, and remain a surprise.
Why?
The students must only get one guess. Otherwise on Monday they can say it'll be on Tuesday, on Tuesday they can say it'll be on Wednesday, etc. This way, the test will never come.
Thus, if they only have one guess, then the test could be on Friday and they might guess Wednesday, or something. Thus, the test would be a surprise since they didn't know when it was held.
Correct me if my logic is wrong. Maybe I'm just rambling, maybe it's already been said, I dunno.
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towr
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Re: Pop Quiz Riddle
« Reply #154 on: Sep 3rd, 2004, 6:18am » |
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The test could be friday only if they made a guess before the end of thursday. Because if the test hasn't been on thursday or earlier, and they haven't made their guess yet, they could 'guess' friday, as it's the only day left, and the test would be cancelled..
If they have to give their one and only guess monday morning before classes start, then it's anyones guess when the test will be.
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Three Hands
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Re: Pop Quiz Riddle
« Reply #155 on: Sep 3rd, 2004, 7:35am » |
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Unless, of course, the students are so convinced that there won't be a test, especially on Friday, that they refuse to believe that the professor will have a test on Friday. Of course, if the students have not guessed and had a test by the end of Thursday, then they have deserve whatever test they have coming...
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Hobbes199
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The "surprsie" should be interpretted as that they will not know it until the day of the quiz. If you can prove each day that it will have to be the next day, then you can avoid the quiz.
If you don't have the quiz on Thursday, you will know it would have to be on Friday, thus invalidating the teacher's promise.
On Wednesday you could use the argument that it can't be on Friday to prove it would have to be on Thursday, but you can only base that knowledge on the fact that you did not have the quiz already.
Using the argument to predict Tuesday that the quiz would have to be on Wednesday fails, because you don't yet know that you wouldn't have had the quiz on Wednesday to say it would have to be on Thursday.
A working solution though, might lie in the fact that the argument does work at all. That is, the teacher may not be able to prove a paradox is incorrect, but if you tried relying on it each day, the teacher can surely surprise you with a quiz.
If we could construct separately valid, though possibly paradoxical answers for each day of the week, then you could avoid taking the quiz.
One solution proposed was to use "hyper-geometric probability" to prove that it had to occur on Wednesday. Assuming that's not just mumbo-jumbo, then you'd need solutions for Monday and Tuesday still, to avoid the quiz.
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Kent
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It is logical to acertain that if the quiz has not been taken by the end of the day Thursday, they would be able to predict that the test is going to happen on Friday; thus canceling the quiz. The students ASSUMED that the professer would not choose Friday and followed this assumption to eliminate each day.
With the information they were given, there is no way to predict on which day the quiz would fall with certainty until the end of the day Thursday AND ONLY IF THE QUIZ WAS NOT GIVEN TO THIS POINT!
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Tarzul
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What's the flaw in the students' thinking?
This is the questoin being asked. The flaw in the thinking of the students was that they looked at a day-day sequential problem backwards. In there case this is a problem since they are assuming the surprise test was not given on a previous day. There statement
"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."
is then only correct if they make it to Thursday and at the end of Thursday they can say for sure the test is on Friday. On Wednesday however they have no such certainty as it will still be a surprise if the test will be the next day or Friday. The element of "surprise" only goes away at end of Thursday. (5 minutes before the end bell on Thursday they could still have the quiz) Since this is true from Sunday night untill Thursday they cannot use a reverse sequential logic to attempt to solve it. Thus since on no previous day could they know if the test was the next day except at the end of Thursday they should have guessed a day from Monday to Thursday and had a 25% chance of getting it right.
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rmsgrey
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Re: Pop Quiz Riddle
« Reply #159 on: Mar 17th, 2005, 7:32am » |
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But if Thursday actually comes along, the students have the following assumptions/facts:
1) The test hasn't been held yet (isn't Mon, Tue, Wed or Thu)
2) The day of the test is Mon, Tue, Wed, Thu or Fri.
3) The day of the test is unknown in advance.
These three axioms produce a contradiction if they are reasoned about by someone who is aware of their own beliefs.
Since 1 would be an observed fact, not an assumption, it can't be false. So either the test will not be a surprise or there is no test. If the test actually does take place on the Friday, the students will be very surprised.
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colin a
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this is what I THINK: i believe that there wasnt a possible way to determine when it was becasue if there was it wouldnt be a SUPRISE TEST! the anwser he was probly looking for was you wont pick a date until we guess and you will guess different. after all, it is logic.
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Djinon
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Re: Pop Quiz Riddle
« Reply #161 on: Apr 19th, 2005, 11:14am » |
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First, since this is a logic class, the students could just take the easy way and figure that next week never comes. No need to guess because it will never be next week. This is a logic class, after all.
Second, the actual flaw in their thinking is that they assume the professor won't plan to administer it Friday because it would be easy for them to guess. They continue with this logic all the way back, but the flaw is that even though the professor would be making it easy for them if it was not administered before Friday it is still possible. This flaw climbs all the way back through their logic. Just because he would be making it easy doesn't mean he can't, and since this prevents them from making the assumption about Friday, there can be no assumptions about any day based on the first.
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Djinon
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Re: Pop Quiz Riddle
« Reply #162 on: May 4th, 2005, 10:21pm » |
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I posted the above around 100 thread views ago. Am I right or something? Need to have some feedback...
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towr
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Re: Pop Quiz Riddle
« Reply #163 on: May 5th, 2005, 2:22am » |
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The real flaw is that the students can't both believe that it's a surprise test and work out when it is.
They can't believe the professor anymore than if he had told them that it was raining and they don't know it. Because after getting and believing that information, they would both know it's raining and know that they don't know it, which is inconsistent.
It may very well be true, but to believe it makes your beliefs inconsistent.
Likewise you can't work out when a surprise test is, because it wouldn't be a surprise test anymore. The obvious solution is to not believe it's necessarily a surprise. But then they can still only work it out if it's thursday and only friday remains.
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Jonathan La Fratta
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this is the answer, its hard to understand but if your willing to think about it you can get it if your smart enough.
He can give it on the last day, there is no way to proove he can't give it on the last day, if he gives it on the last day he's a liar, if he doesn't give it on the last day, hes a liar.... we know he is not a liar, but if he were it would be impossible to know if he were going to lie about giving it all toghether, or lieng about giving the test , the students might think they know its that day, but to know ahead of time that he will is impossible , the contriditrary statements create an uncertanty which makes it possible to give the test and it be a suprize
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paul schmitz
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the problem with the student's reasoning is that at the end of their logical sequence, they do not guess any day for it to be. therefore, the teacher is able to hold it on any day, since the students did not guess at all.
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Jujubeeks
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The problem with the students reasoning is that it would be "too obvious". I assume they can't presume the day on the day of the test, so then unless the test was on a Friday, they couldn't guess the day. The test could be on any day besides friday. They assume it would be too obvious, but the professor is obviously a genious using reverse psychology. It would be obvious, but that doesn't mean it would be impossible. If that doens't make sense, then figure this: The test could be on a tuesday because on monday, you couldn't be able to tell which day it was. It could still be wednesday or tuesday. The students rule them as impossible because it would be too obvious. However they are both as "obvious" as eachother and therefore just as likely as eachother.
Plus what does the professor care? He could reassign them the test next week. That'd be a real surpise...
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Brad Bida
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Seeing as it was told in a logic class...
I think that their flaw was assuming one reasonable deduction, (friday would not be the test because they could guess it on thursday and be correct), on the entire problem.
As for figuring out what day the test is on - it's beyond me!
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Nasta
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Re: Pop Quiz Riddle
« Reply #168 on: Sep 3rd, 2005, 7:24pm » |
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a little clarification on the problem please - when do the students have to guess? how many times can they guess? do they receive an answer whether they're right? (these in the case that they indeed pick a day, and not say there won't be a test) can they guess every day for the rest(e.g. on monday say tuesday or thursday, then on tuesday for wednesday or friday and so on) and again - do they receive an answer? the problem itself suggests something like this - if it's thursday after classes then we'll say to the professor it's tomorrow and he'll cancel, but have they not tried before during this week(i mean i realize this doesnt happen because of their "logic", but in the supposed situation what would be the case)? another thing - is it possible that the professor anticipated their answer and it's logic and by not denying it he made them believe that they're correct and thus the test was truly a surprise(i mean he planned this whole thing, so that they hang themselves) or this is not relevant? by his cagey answer i think this was indeed his purpose, although if he hadn't told them at all the test would also be a true surprise, and in the former case there was still possibility that they come up somehow(flipping a coin for example or whatever) with whatever day and thus 0.2 probability of cancelling the test altogeter and otherwise spoiling the surprise anyway. cancel or not, they would have expected it to be this week. hmmm. i am confused.
with some assumptions made i agree that friday would be known after thursday and won;t be much of a surprise. but after wednesday they cannot rule out both thursday and friday, should they take their guess then(i assume only one, one-time guess, no answer from professor). they can go(on wednesday after classes) with thursday of course - if it was planned for then, it would be cancelled, if not - it would be on friday. but on tuesday night, should they decide to take a guess then, it's not so easy. they should either pick wednesday or thursday, and if they are correct it would be cancelled, if not that leaves 2 days that can be "surprising". their logic fails to acknowledge that escalading of the possible days the test can be. they just guess that if it hadn't been on friday because it wouldnt be much of a surprise, then the professor wouldn;t make it on thursday, because they would know it my wednesday and so on...but that's not true. that's their guess and not the real world situation. moreover this somehow makes assumption over assumption and all of this based on the assumptions that we can choose any day to take our guess. but if they guess on monday(again - when SHOULD they guess)? then what? even friday remains a surprise because if they chose tuesday for example and it was really planned for then, then on thursday they would either have to think it's friday or that it;s cancelled, and they cannot tell which one. so they cannot rule out friday even on thursday. so it is really important to specify when they should in principle take a guess and how often and so on...
again ...
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| « Last Edit: Sep 3rd, 2005, 7:47pm by Nasta » |
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dhasenan
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Here's a thought.
The professor knew how the students would think. They realized he wouldn't be able to give a test on any day, then concluded that he would not. Therefore it was a surprise when he did. And he taught the students something important: logic is good, but sometimes it's better to be clever.
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rmsgrey
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Re: Pop Quiz Riddle
« Reply #170 on: Oct 30th, 2005, 5:11am » |
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on Oct 29th, 2005, 7:36pm, dhasenan wrote:Here's a thought.
The professor knew how the students would think. They realized he wouldn't be able to give a test on any day, then concluded that he would not. Therefore it was a surprise when he did. And he taught the students something important: logic is good, but sometimes it's better to be clever. |
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Except that logic would still have worked if the students had got it right - as explained earlier in the thread, they get a contradiction and assume it comes from only one of their assumptions, ignoring their other assumptions.
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nnahas
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Re: Pop Quiz Riddle
« Reply #171 on: Nov 18th, 2005, 1:03pm » |
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I have read some of the posts in this thread, and I still haven't made up my mind on which point of view, if any, is correct. I vaguely think the problem has to do with the word "predict" and "surprise" being ambiguous. But it is clear that the flaw in the students thinking, if there is one, is very subtle, because their reasoning correctly proves that there is no computer algorithm that can provably guarantee that it will always output a day of week that can't be predicted , provided the algorithm code is known to the students, even if the algorithm has access to a truly random source.
I am not sure if the assumption of knowing the algorithm beforehand can be weakened.
Now, here is another riddle, and I don't know the correct answer.
Suppose that someone proves that beal's conjecture(http://www.math.unt.edu/~mauldin/beal.html) is undecidable. Then it has no counterexample,then
it is provably true, then it is not undecidable.
so assuming that this problem is decidable leads to a contradiction, therefore it must be decidable.
the question is : Is the above reasoning correct or not?
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Icarus
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Re: Pop Quiz Riddle
« Reply #172 on: Nov 18th, 2005, 7:42pm » |
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No - it is far from correct!
If it is undecidable, then it is undecidable. Having no counter-example does not imply that it is provable! If it is undecidable, then its negation (i.e., the statement that the conjecture is false) also has no counter-example. So by the same reasoning, it would also be provably true. And so by this reasoning, to have an undecidable statement in a mathematical theory would make the theory contradictory.
Yet Godel proved that any mathematical theory sophisticated enough to include the natural numbers must have undecidable statements. If your reasoning were true, the existance of natural numbers is contradictory. 
Instead, we note that "provable" means that you can deduce the statement logically from the axioms of the theory. Just because a statement has no counter-examples does not imply that this is possible.
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towr
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Re: Pop Quiz Riddle
« Reply #173 on: Nov 19th, 2005, 3:22pm » |
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on Nov 18th, 2005, 7:42pm, Icarus wrote:| Having no counter-example does not imply that it is provable! If it is undecidable, then its negation (i.e., the statement that the conjecture is false) also has no counter-example. |
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I don't think that's quite right. If a statement has no counter example it must be true, and so it's negation must be false and thus have a counter example.
However, if the statement is undecidable, you can't know if the statement does or doesn't have a counter example. (Because that decides it)
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Eigenray
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Re: Pop Quiz Riddle
« Reply #174 on: Nov 19th, 2005, 4:41pm » |
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The only thing one might call a "counter-example to the negation of Beal's conjecture" would be a proof of Beal's conjecture. But it's a bit odd terminologically as "counter-example" is usually reserved for disproving universal (proving existence) statements (although even if a universal statement has a counter-example, that doesn't mean it's provably false).
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| « Last Edit: Nov 19th, 2005, 4:44pm by Eigenray » |
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