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Grimbal
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Re: Pop Quiz Riddle  
« Reply #125 on: Apr 27th, 2004, 1:53pm »
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I think the professor did not lie, but bluffed the students because he could not tell reliably what the student's conclusion would be.
 
As mentionned earlier, the set of axioms is inconsistent.  In that situation, every statement is true, and also the opposite of every statement.  For instance, the students conclude logically that it can be only on Monday, then that it can not be on Monday.  Then they conclude that there is no test, even though an axiom says there is one.
 
The students came to the conclusion that there would be no test.  With a few different twists, they could have come to the conclusion that the test is on Tuesday (once they know it can not be on Monday, with an older fact that it is not on Wednesday to Friday, they could decide it must be on Tuesday).
 
My conclusion is that the professor was correct by chance rather than by his being truthful and reliable.  His claim was not certain.  He just bluffed the students and won his bet.
 
So the flaw in the student's reasoning is that they trust the professor only to make claims of which he is 100% sure that they will not be proven false.  Once the students accept that the professor is not 100% reliable in his claim, they can not conclude for sure that the test can not be on Friday.
 
Consider what happens if the test is on Friday.  On Thursday night, they will come to the conclusion that the test is on Friday, and so, it is not a surprise any more.  So the professor lied.  But once they accept that the professor might lie, they can not be sure there will be a test at all.  So, there could be a test on Friday, even though they won't know for sure.
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Re: Pop Quiz Riddle  
« Reply #126 on: May 19th, 2004, 3:57pm »
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The question is: why can the professor give the test on Tuesday, even though the students determine that it won't be held at all?
 
He said it would be a surpise, and he would cancel it if anyone could figure out what day it would be. This obviously means the students must tell him the day before the test (if not at the exact moment he posed the riddle for them).  
 
Moreover, like most people point out, it couldn't be Friday, because Thursday would come around, there wouldn't be a test, and everyone would know it was tomorrow (key point: as long as they didn't guess incorrectly already!). Keep in mind the professor would also figure this logic out, and not have the test on Friday (given that they haven't guessed incorrectly yet).
 
It continues, "The test couldn't be on Thursday, because Wednesday would come around, there would be no test, and the students would know it was on Thursday, since it can't be on Friday". Hence it's not a surprise still...
 
This is the point where the continuity falls apart. The professor could know the students are thinking this way, that the test must be on Thursday if it isn't on Wednesday, and hold the test on Friday.
 
The logic seems sort of strange, but think of it like the professor knows logically that it would be unlogical and, therfore, improbable he would hold the test on Friday, so he does so anyway. If it's improbable, wouldn't that be the biggest surprise?
 
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Re: Pop Quiz Riddle  
« Reply #127 on: May 19th, 2004, 4:14pm »
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I think that the flaw is in thinking that they can logically determine which day it will be on, since the professor knows that they have to pick one day for the test to be on in order to cancel it. Given the thinking of the class, they must always consider that the test will be that day, given that it would not be a surprise if they were given it on the next day, because they would be expecting it on that day. Essentially, the students are expecting the test on every single day, but because they expect the test every day, they then consider that it would not be a surprise test if it occured on any day.  
 
Also, by telling the class there will be a test, the professor lies by saying "there will be a surprise test next week", unless the students believe that there is not going to be a test. The flaw in the student's thinking is inthinking the professor must be telling the truth, since the sentence is contradictory. If they believe there will be a test, then it will not be a surprise (or, at least, not much of a surprise) whenever it happens, but if the students don't believe there will be a test, then they will be surprised by the test. So the flaw in the students' reasoning is believing that there will be no test the next week, instead of expecting a test every single day, given that the student's reasoning only works if they are anticipating the test for the next day.
 
Unless, of course, the professor sets two tests - the second one being the surprise...
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Re: Pop Quiz Riddle  
« Reply #128 on: May 20th, 2004, 3:39am »
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The fun thing is that the professor's statement turns out not to be a contradiction - it in fact turns out to be the exact truth - though there are certainly possible scenarios where it could turn out to be false.
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Re: Pop Quiz Riddle  
« Reply #129 on: May 20th, 2004, 5:50am »
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Urgh!
 
If you are so effing sure that it can't be on Friday then I won't cancel the test.
 
(Its on Friday).
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Re: Pop Quiz Riddle  
« Reply #130 on: May 20th, 2004, 5:58am »
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Sorry to double post but just to elaborate...
the point is, the students aren't allowed to guess EVERY day like that. That is the flaw in their reasoning. If they assume it can't be Friday then Friday it might be. If they guess it will be cancelled then (SURPRISE!) it could be Tuesday or any other day.  
 
The real question is... how can I cancel the test at all? since if  I  said it was cancelled you may as well go to the uni bar to "study" with your homies. SURPRISE! the test is on after all. Like I said, it was a surprise mofos; I cheated. hahahahha
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Re: Pop Quiz Riddle  
« Reply #131 on: May 20th, 2004, 10:31am »
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Pretty much as I said - the best the students can do is say "the test cannot occur on that day, because I would be expecting it on that day, since it couldn't be the next day, because I would be expecting, etc.". Since they only get one guess at when the test is, they could only eliminate one day. However, their logic assumes that they are expecting every day after today, so it must be today, but that means it can't be today. Essentially, they go from expecting every day (which is what they should do, and possibly guess when the test is planned for a 1/5 chance of cancelling it if the professor is playing fair) to expecting the test never to happen - which is where they make a mistake...
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Re: Pop Quiz Riddle  
« Reply #132 on: Jul 4th, 2004, 3:13pm »
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the flaw is that they think MONDAY and FRIDAY are the same.
 
THEY ARE NOT
 
Based on the half correct logic they use,
 
Friday is the least likely day.
 
AND Monday is the most likely day.
 
so the professor gives it on tuesday (because he knows this)
 
because wednesday would be the middle of the week  
 (the hump in the curve)
 he does not risk giving his student a greater chance to guess
 
As days are eliminated the probablity of guessing correct becomes higher  
 
So why not give the quiz on the day  
right after the most probable day
 
In fact a smart student would know that the only days he can give the quiz with out someone guessin correctly is tuesday or wednes day
 
Then by more reasoning a smarter student knows it cannot be wednesday  
 
IT MUST BE TUESDAY
 
(NOTE: This only works when the professot himself knows when it is)
 
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Re: Pop Quiz Riddle  
« Reply #133 on: Jul 5th, 2004, 9:47pm »
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The way I see it is that they have eliminated all the days in their student logic, so they are now certain in their undying belief that there will not be a popquiz. So after monday with no popquiz they are sure there won't be one tommarrow because there can't be on Friday or Thursday or Wedsday or Tuesday, since they don't see it happening it is a surprise  Roll Eyes  
Or solution 2 the class consists of gold fish and so they can't remeber that he told them there was going to be a pop quiz  Tongue
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Re: Pop Quiz Riddle  
« Reply #134 on: Jul 6th, 2004, 6:52am »
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In response to GUEST's post, I do not see why the professor is so limited in his choice of days for setting the test to just Tuesday. Essentially, all he needs to do is out-bluff the students. He could, indeed, have the test on Friday, since it would be a surprise to he class that the professor had the audacity to set the test on the day that they would be certain they were having the test, so long as they trust what the professor says. The flaw in the reasoning, I believe, is that the students come to a conclusion which contradicts the statement made by the professor, and so, because they believe that the test can never be a surprise, they believe that there is no test, which then makes any test a surprise. Hence, the students fulfil the professor's statement only through not believing there will be a test - if they believe there will be a test, then they will not be surprised by any test given, but cannot accurately predict when the test will be. The cunning part of the professor's statement is in stating that the surprise test will happen unless they can tell him when it will be, making the class infer that there must be some means of predicting accurately when the test will be.
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Re: Pop Quiz Riddle  
« Reply #135 on: Jul 6th, 2004, 5:42pm »
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The students' solution to this puzzle relies is derived along a very particular path.  In other words, each deduction is based entirely on the previous deduction, such that the reasoning which eliminates Thursday is based on the previous reasoning which eliminated Friday, and so on.  As long as each of these deductions proves true in practice, the ultimate conclusion that there will be no test is accurate.
 
However, the teacher has the advantage that if he breaks from the traditional deductive reasoning which brought the students to their conclusion, he will leave them without a guess.  While the deduction that - assuming there is a 100% chance of a test during the week - Friday will not be the day of the test remains true regardless of our reasoning, it is only one level deep.
 
But based on the student's logic, even Monday is ruled out, thus denying the teacher the ability to surprise.  The minute the teacher chooses a day he is disolving the logic the students used to prevent him from doing so.
 
For example, if the Teacher chose Monday and the students came to him with their very narrow logical explanation - that there would be no test - then he could simply say, "no - sorry you're wrong, I HAVE chosen a day."  The students, if given another chance, would walk away, without the freedom to choose the same answer again... and five days around which to make a prediction.
 
They could still deduce that a test on Friday would be Just-in-time predictable.  But they could not deduce that a test would be JIT predictable on Monday or Tuesday, for example.  Their deductions ruled out every day despite the fact that the teacher had chosen a real day.
 
The students could say on Monday, "Well there wasn't supposed to be a test today because we deduced that it was impossible.  But since it's not impossible it must be today, or tomorrow, or the next day."  The teacher, therefore, could have it on Monday, Tuesday or on any day that isn't Friday, and in each case the students would not be 100% sure.
 
Based on this reasoning, the students could argue that they would expect the test EVERYDAY based on their original reasoning only to be proven wrong at the end of the day at which point they would focus on the following day.  Thus the test would never be a surprise in that it is always expected thereby forcing the teacher to cancel the test.  That is assuming the students are not limited in their number of guesses.
 
In summary - the teacher has the advantage because he can destroy the students' compound premise by choosing a day and contradicting their conclusion - that it can't be on any day.  Once this premise is disturbed, it no longer holds.
« Last Edit: Aug 5th, 2004, 6:11am by mattian » IP Logged
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Re: Pop Quiz Riddle  
« Reply #136 on: Jul 6th, 2004, 5:50pm »
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More simply:
 
If the teacher chooses a day - Say Tuesday - and the students are given the challenge of guessing, then suppose the students give their solution - that there would be no test - and it's wrong (because the teacher chose Tuesday).  The teacher would give them another chance - but he would allow them to make the prediction just-in-time (ie. on the morning of each day).
 
So on the Monday morning he would say the students, "Is your test today?"  And the students would not be able to answer with 100% certainty.  Even on Thursday morning, they would not know the answer.  Friday morning is the only morning that they could give a certain answer of "Yes".
 
This proves that the teacher could choose Friday (as long as the students have only one guess on the morning of the test).  Because he would ask them on Thursday, and they would not be able to say if the test were to be given on Thursday or Friday.  They might deduce that Thursday would make sense but they would get it wrong and the deal would be off by the time the test came around on Friday.
 
« Last Edit: Aug 5th, 2004, 6:04am by mattian » IP Logged
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Re: Pop Quiz Riddle  
« Reply #137 on: Jul 30th, 2004, 8:58pm »
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I scanned through the entire thread to make sure what I will say has not been said, and I didn't find it.
 
Keep in mind the riddle didn't ask you to guess what day, or why the professor chose tuesday, but what the flaw in the students' thinking was. Immediately, guessing "there will be no pop quiz" is pretty useless, but logical. Yep, logical. The day of the pop quiz had already been selected by the professor. Using logic, he predicted that their answer would be "no pop quiz", and therefore it was a surprise to them in the end. In fact, unless they guess the day correctly, it will always be a surprise quiz.
 
The flaw in logic for the students was that their own teacher would not have predicted their answer.
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Re: Pop Quiz Riddle  
« Reply #138 on: Aug 5th, 2004, 12:31am »
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Here's how I see it. The professor picked Tuesday for no other reason than to fake out the students. Here's the logic. He know his students, being the geeks that they are, will study all weekend and be prepared for the "surprise" quiz on Monday. They in turn will be let down when they don't get the quiz on Monday and will be some what caught  offguard on Tuesday. So the Professors logic in giving it on Tuesday is sound. The flaw in the students thinking is that the statement has been made by a Teacher of Logic and that by saying there is a Pop Quiz nullifies the surprise element therefore coming to the conclusion that there is no pop quiz. So in a nutshell, the professor used Logic to fake out his students.
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Re: Pop Quiz Riddle  
« Reply #139 on: Aug 5th, 2004, 2:37pm »
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I think ultimately the seven-day problem can be reduced down to the single-day problem. Imagine a professor who simply states, "There will be a surprise test tomorrow."  
 
Any student who believes that there is a test won't be surprised, so the professor's statement is true iff the students believe it to be false.  
 
It seems that this ultimately becomes a case of the professor saying, "This statement is true only if you believe it to be false." In other words, nonsense equivalent to "This statement is false."
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Re: Pop Quiz Riddle  
« Reply #140 on: Aug 6th, 2004, 5:34am »
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on Aug 5th, 2004, 2:37pm, guest wrote:
It seems that this ultimately becomes a case of the professor saying, "This statement is true only if you believe it to be false." In other words, nonsense equivalent to "This statement is false."

 
{this}=({this}=B({this}=F)) is not the same as {this}=({this}=F). The former simplifies to T=B({this}=F) while the latter simplifies to T=F (in both cases using associativity of identity followed by (X=X) being an axiom)
 
The conclusion drawn from the first statement is "The statement is true and I believe it to be false" Only if you believe that you are infallible do you actually believe the statement to be false (and thus believe a contradiction), and only if you are aware of your own beliefs (that is if you believe something you believe that you believe it) do you believe that you both believe the statement true and believe it false at which point you are certainly inaccurate, but not necessarily trapped into believing a contradiction...
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Re: Pop Quiz Riddle  
« Reply #141 on: Aug 6th, 2004, 7:24am »
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Sheesh!
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Re: Pop Quiz Riddle  
« Reply #142 on: Aug 7th, 2004, 12:33pm »
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Sorry, my mathematical training takes over sometimes, particularly when dealing with "fun" stuff like symbolic logic (OK, so I have a warped sense of what's fun...)
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Re: Pop Quiz Riddle  
« Reply #143 on: Aug 7th, 2004, 12:35pm »
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It wasn't the mathematics that baffled me - it was the English!
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Re: Pop Quiz Riddle  
« Reply #144 on: Aug 20th, 2004, 11:30am »
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There are too many responses for me to read, so maybe this has already been addressed, but here are a couple of thoughts:
 
I think it is funny that the students conclude there is no test because they believe what the teacher said is true while their conclusion would imply that the teacher was a liar in the first place.
 
Think about the 2 day simplification:
The professor wants to pick a distribution (1-p, p) for giving the test each day such that the odds of the students predicting the test with full confidence is minimal. Therefore, he wants to pick p as low as possible. However, p can't be 0 because then the students would know it was on the first day.
However, because the students make this deduction, if the professor switches p to 0, he will fool them all the time, but the students, knowing the professor knows how they think will anticipate this, thus the professor must set p > 0 to have better than a 0 fooling rate, thus because the students are so smart, the professor can set p = 0.......
 
Reminds me of that scene in The Princess Bride.
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Re: Pop Quiz Riddle  
« Reply #145 on: Aug 26th, 2004, 9:35am »
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Wow. You're all making this much more complicated than it really is, just like the students did. You're all making the same mistake. No one liked. No one bluffed. Reread what you're given. The professor made three statements:
 
Quote:
We're going to have a surprise quiz next week

 
Okay. Simple enough. The semantic game is silly, since it's already not a surprise, we were just told about it.
 
Quote:
but I'm not telling you what day

 
Okay, the day is not being communicated. Fair enough.
 
Quote:
if you can figure out what day it will be on, I'll cancel the quiz

 
Here's where the flaw is made. The students, and most posters, have assumed that the teacher doesn't want anyone to be able to guess the answer. He never said that. Most riddles we are told the pirate wants as much money as possible, or as much bloodshed as possible, or something to give his decision-making direction. In this case, no motivation is given. The teacher has no stated interest in keeping the students from guessing the day of the test. The flaw in the students' logic is the assumption that the teacher has a horse in the race, which he doesn't. He has no particular interest in whether the test is given or not. So there was no logic in determining what day he would give the riddle. It could have been Friday. The students could then guess Friday, and the test would have been cancelled. Simple.
 
However, what I find interesting it the nature of the situation if the teacher did have motivation to give the test. Assuming the semantics that a "surprise" test must truly be a surprise, the very act of eliminating a day from consideration actually places it back into consideration. By determining that the test can't be given on Friday, the students make Friday the most surprising day on which to give the test. So, having eliminated the entire week and concluding that no test would be given, they actually insured that the test could be given on any day and even increased the element of surprise which had been taken away by informing them that there would be a surprise test during the week.
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Re: Pop Quiz Riddle  
« Reply #146 on: Aug 27th, 2004, 12:59am »
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on Aug 26th, 2004, 9:35am, daniducci wrote:
Okay. Simple enough. The semantic game is silly, since it's already not a surprise, we were just told about it.
Sure it's a surprise, because you don't know when it would be given. That's the surprising element of it.
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Re: Pop Quiz Riddle  
« Reply #147 on: Sep 1st, 2004, 12:50pm »
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*(assuming the professer will only give one quiz in the week and this is known by all)
 
THE STUDENTS:
There are two flaws in the students thinking.
 
    The First one was not regarding the professor's statement as a truth.  He had said 'there will be a surprise quiz unless you guess the correct day'.  The only way to cancel the quiz was to guess the correct day.  They did not even guess a day, so the quiz had to happen.
 
    The second was the assumption that there can not be a surprise quiz on Thursday.  This led them to the false conclusion that there can not be a surprise quiz at all in the week.  
    Obviously there can't be a 'surprise' quiz on Friday because after Monday-Thursday pass, Friday is the only remaining day, and the quiz at this point is no surprise.  However this does not mean that Thursday is necessarily out.  
    After Monday-Wednesday have passed, the students could not assume that the quiz is definately not on Friday because there is no way to know the precise thinking of the professor. So a quiz on Thursday would be a surprise since they can not be 100% certain that the professor's thinking wasn't flawed. (or maybe he's just being sneeky and trying to outsmart them. who knows) Think about it as if the week were 100 days long.  Day 100 is the only on that a quiz could not be a surprise with 100% certainty.
 
THE PROFESSOR:
    It has not been considered in this thread what the affects of the student's guess must have on the professor's thinking in picking the day.  
    If the students guess a wrong day, that narrows down one day, and gives them more information about the week, thus changing the surprise factor.  For this reason, and this reason only, some days of the week other than Friday are unable to, with 100% certainty, be surprise quiz days.  If the professor did not allow them to guess a day to cancel the quiz, and just said there will be a pop quiz next week, the only day it could not occur on would be Friday.  However the professor does give them a guess, and this makes his day picking a little more tricky.  
    He must pick a day that, regardless of what day his students guess and what logic they use, will not be able to be determined with 100% certainty to be the quiz day.  Otherwise it's not a surprise.
 
    The professor is obligated, by his word, to have a surprise quiz the next week.  Friday can't be the day he chooses because being the last day, it could not be a surprise.  If he chooses Friday, it is a certainty that there would not be a surprise quiz the next week. To be sure of his pick he must assume his students see this obvious logic also.  So he considers Thursday.
 
Thursday, with no student guess, could be a surprise. (as stated above)  However, IF the day the students guess is Friday, Friday will then be ruled out with 100% certainty After Monday-Wednesday pass Thursday would not be a surprise.  Because the professor is giving the students a guess, if he chose Thursday, it is possible that it would not be a surprise.  So he considers Wednesday.
 
Wednesday will not work for a similar reason. If the students guessed Thursday it must be ruled out as a possible quiz day. To INSURE a surprise the professor must be safe and ASSUME that the students will rule out Friday using the same logic that he did. Then after Monday and Tuesday have passed, he must assume that they know Thursay-Friday are out, so a quiz on Wednesday would not be a surprise.  So he considers Tuesday.
 
Tuesday will work.  A quiz on Tuesday would be a surprise. No single guess by the students would allow them to logically rule out all days following Tuesday in the week.  
 
So the professor chose Tuesday.
 
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Re: Pop Quiz Riddle  
« Reply #148 on: Sep 1st, 2004, 2:56pm »
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So the students are convinced, 100% that there will not be a quiz on Friday? Then if I were the professor, guess which day I would pick for it. I'm reasonably certain that the students would be surprised to be given the test on Friday.
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Re: Pop Quiz Riddle  
« Reply #149 on: Sep 1st, 2004, 10:52pm »
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I never said that the students were convinced that there was not going to be a test on Friday.
 
I said that a test given on Friday could not possibly be a surprise.  This should be obvious to all parties but the students can never be sure which day the prof. has picked.  
 
The professor has a responsibility and must assume the students have ruled out Friday in order to make the statement 'there will be a surprise quiz' true.
« Last Edit: Sep 1st, 2004, 11:01pm by honkyboy » IP Logged
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