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riddles >> hard >> Pop Quiz Riddle
(Message started by: Jackass on Jul 30th, 2002, 5:25am)

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Title: Pop Quiz Riddle
Post by Jackass on Jul 30th, 2002, 5:25am
anyone know the answer to the pop quiz riddle? ??

Because I don't.  :(

//Question moved from title, title changed by Icarus

Title: Re: anyone know the answer to the pop quiz riddle?
Post by pio on Jul 30th, 2002, 5:33am
Well, I don't know if this is the right answer, but to me the students premise is false:

"It can't be in friday because then we would know it on thursday"

And? What's the problem with knowing it when there's none left? It can perfectly be on friday. They would find it out on thursday, ok. But that's not a premise to base the rest of the reasoning.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 30th, 2002, 7:32am
Because on friday it won't be a surprise test anymore, the professor said it would be a _surprise_ test. He can't hold it on friday since the students will not be surprised, rendering his premise it would be a surprise test false.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Neil Sedaka on Jul 30th, 2002, 8:05am
Which is why leaving it until Friday would be THE BIGGEST surprise possible!
Pity the students as they studied each night up to and including Wednesday night, each time increasingly sure that the test would be the next day.
The surprise actually hits them on Thursday!

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 30th, 2002, 8:37am
You two bring up an interesting paradox. If the teacher said "You won't be able to figure out when it is" meaning it will be a TOTAL surprise, the students logic would work. But if the test isn't gone until Thursday (the students are at home on wednesday night) they won't know when it is. It could be on thursday, or on friday.

Logic if the popquiz is a TOTAL surprise (meaning they will never know when it is, until they get the pop quiz)

If it is thursday night, the pop quiz must be on friday and it wouldn't be a surprise (because friday is the only day left)

If it was wednesday night, the pop quiz must be on thursday. If it was on friday, it wouldn't be a surprise (because friday was the only day left). Since it can only be on one day that would be a surprise at all, it would be known when it must be, and so it wouldn't be a surprise.

If it was tuesday night, the students could rule out friday, and thursday (because it wouldnt be a surprise as listed above) and so wednesday is the only day left to have the quiz. Since only one day can be had for the quiz, it isn't a surprise.

If it was monday night,  the students could rule out friday, thursday, and wednesday. So Tuesday is the only day left, and it wouldn't be a surprise.

Similar logic can be used... see?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 30th, 2002, 8:38am
Still if thursday passed without test, the test is no _surprise test_ anymore, due to this he canot be held as such.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 30th, 2002, 9:00am
Are you agreeing with me?  ;)

If the teacher means "I am planning to give a test this week. If you know when it is, and you are right, I will cancel it." Then it can be on anyday but friday.

If it was wednesday night, they wouldn't know when it was. Here is what I mean:

The test has no problem with Friday. It won't happen, because the students can cancel it, but that still doesn't stop them from planning the test.  If the teacher says "I *WILL* give a test this week" instead of "I am planning to" then it would be the same as "it *WILL* be a secret (See previous post)"

If it was wednesday night, they couldn't say if it was thursday or friday. If it was thursday, they couldn't rule out friday, because it can still be planned then, and since that makes *2* days not *1* day. I hope this makes sense!

Similar logic can be applied to the other days... see?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 30th, 2002, 9:58am
Yes, Well I replied to the post before yours, yours got in between, and well I agree with that.

The problem here is that this riddle here is a bit modified, invalidly IMHO, disquising the orignal paradox. In this case the students suppose the teacher is mean, and don't want them to get away with a test, however that is not a must be. They can't say if the professor wants them to guess the day friday right.

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.

Now what's wrong with logic?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Alex on Jul 30th, 2002, 2:18pm
Take, for example, that it is Tuesday night. You declare that it cannot be on Friday because that wouldn't be a surprise as of Thursday night. Since it cannot be on Friday, then on Wednesday night it would be obvious that it is on Thursday, because Friday would be obvious. Therefore, since having the test on Thursday would be too obvious, it must be on Wednesday, right?

Prediction is assumed in this scenario. You are assuming that the teacher will decide to give the quiz on the most unpredictable day, when the riddle says nothing of this. The quiz will simply be on a random day. It might be Friday. It might be Monday.

I truly have no idea how to solve this riddle.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Jul 30th, 2002, 3:03pm
Let me contribute my thoughts on this problem. I think it's a correct solution, but unlike problems with numerical answers, it's difficult to empirically determine when an answer is "correct".

First, as I read the problem, the teacher announces that there will be a surprise quiz. The meaning of this is "on a given night, the students cannot deduce whether or not there will be a quiz the following day". If they can deduce the day the quiz was planned for, then the quiz will be cancelled. Note that the planning of a quiz and the actual quiz taking place are not the same thing; it can be planned for a day but then cancelled because the students were able to deduce the day correctly.

The tricky part of the problem is that it attempts to create logical propositions without attaching required context. Specifically, the logical propositions are valid only at certain times. Let's restate the first line of the problem logic with this implicit assumption spelled out:

1) If the quiz didn't happen on Thursday and it's Thursday evening, then the quiz is on Friday.

We cannot know before Thursday evening that the quiz did not occur on Thursday. We are not told that the quiz isn't Thursday, so the only way we can find out is to wait for Thursday to pass. Once we reach Thursday evening with no quiz, then we can deduce that the quiz is on Friday.

Now, if we revisit the problem, the rest of the logic is moot. It all relies on the assumption that it can't be Friday since we'd be able to deduce that it is Friday. However, this proposition only becomes true once it is Thursday evening. At any time before, we cannot satisfy the prerequisites of proposition 1.

Thus, at any time before the end of class on Thursday, the quiz can be any day. Thus the quiz can be on Tuesday without violating any rules.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 30th, 2002, 8:28pm
Yes, what Nicodemus said above is what happens when the teacher doesn't make it SURPRISE all the way. So, it is a good explanation of the problem! Good job...

SURPRISE all the way means the students will never know when the test is, until it is handed out.

Alex, nobody ever said it the teacher favored over days, simply he can't choose from days that aren't a SURPRISE all the way If the quiz is SURPRISE all the way (like the kids thought) then it can't be any day.

Logic if the popquiz is a TOTAL surprise (meaning they will never know when it is, until they get the pop quiz)

If it is thursday night, the pop quiz must be on friday and it wouldn't be a surprise (because friday is the only day left)

If it was wednesday night, the pop quiz must be on thursday. If it was on friday, it wouldn't be a surprise (because friday was the only day left). Since it can only be on one day that would be a surprise at all, it would be known when it must be, and so it wouldn't be a surprise.

If it was tuesday night, the students could rule out friday, and thursday (because it wouldnt be a surprise as listed above) and so wednesday is the only day left to have the quiz. Since only one day can be had for the quiz, it isn't a surprise.

If it was monday night,  the students could rule out friday, thursday, and wednesday. So Tuesday is the only day left, and it wouldn't be a surprise.

(If it was sunday night, Monday would be the only day left, other than tuesday, wednesday, thursday, and friday)

Note that if his meaning of a "surprise quiz" is a "SURPRISE all the way quiz" then he lied. But if it isn't, he didn't... To summarize Nicodemus' ideas:

You can't rule something out that hasn't happened yet, if you can't be sure it is happening.

Also, my other idea from one of my first posts:

The test has no problem with Friday. It won't happen, because the students can cancel it, but that still doesn't stop them from planning the test.  If the teacher says "I *WILL* give a test this week" instead of "I am planning to" then it would be the same as "it *WILL* be a secret (See previous post)"

If it was wednesday night, they couldn't say if it was thursday or friday. If it was thursday, they couldn't rule out friday, because it can still be planned then, and since that makes *2* days not *1* day. I hope this makes sense!

Do you understand?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Jul 30th, 2002, 9:50pm
Gamer, I think I see the distinction you're talking about. Let me see if I can restate it:

We can use a stricter definition of surprise meaning that the students won't know when the test is, even the night before. This slight difference in meaning has major implications: If the professor is limited to picking a day that will be a strict surprise, then he cannot choose Friday because at some time before Friday it becomes predictable. Specifically, Thursday evening it becomes obvious the test is Friday.

Thus we eliminate Friday completely. Our strict definition removes the temporal prerequisite from the proposition by disallowing days that will ever become predictable.

My other argument hinged on the ability to discard the unconditional conclusion "it can't be Friday". Under the strict surprise definition, this is now a valid conclusion. Therefore we can follow the logical chain of the reasoning put forth by the students... Each successive day uses the same logical argument as the available days in the week diminish. Eventually, there are no days remaining, so there cannot be a test.

Gamer, I hope I summarized your argument correctly?

If so, we can at last explain the problem (in mind-numbing detail). The flaw in the students' reasoning (since evidently there was a quiz) was in their interpretation of the word "surprise". They followed the above chain of logic from that premise, while the professor followed the logic I and others detailed before.

Now Mr. Wu can remove the "unsolved" mark on this problem! :)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 30th, 2002, 10:44pm
No he can't. Who said the professor followed your or the others logic? He can as well follow the students logic, saying the quiz will be a surprise on the day it's handed out. It was a surprise just then, wasn't it?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Jul 31st, 2002, 12:10am

Quote:
 He can as well follow the students logic, saying the quiz will be a surprise on the day it's handed out. It was a surprise just then, wasn't it?

That sounds like the solution we arrived at. If he was following the student's logic, there cannot be a quiz because it is a logical contradiction (between there being a quiz and it being a surprise). If he does give a quiz anyway (as you say and as stated in the problem), then it must be because he is using a different interpretation of "surprise" in his original statement, as explained above. (Or he flat out lied, but we're overlooking that.)

So where does your example disagree with this solution?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 31st, 2002, 1:03am
I refrain from putting this thing back to shelf too fast, we may be on the right track, but doesn't mean we solved it.

"then it must be because he is using a different interpretation of "surprise" in his original statement, as explained above. (Or he flat out lied, but we're overlooking that.) "

So thats okay let's look on the professor. Can we really overlook that he flat out lied? Maybe he just did.

I think a key on understanding the true problem is to modify the number of dates the professor anounces that a surprise test will be possible. Let's call it n. So in the original puzzle n is 5. I think the size of n doesn't actually matter to the problem. Now what happens if n would be 2, there is a suprise test tomorrow or the day after tomorrow. Or let's go even more extreme, let's make n 1. "The will be a surprise test tomorrow." Huh? His argument is just invalid, false. However the students consider is argument to be false, and are surprised by the test, so the argument was actually valid in this case.

So I think this neares the core of the problem. Okay you're solution is bailing out of this paradox, since you suppose the professor wanted to say a valid argument, and was only interpreted false, or just mismatched words a bit. You suppose he didn't want to lie or tell a paradox. But suppose he just did. His argument became valid after he dealt out the test, so he didn't lie at the end.

I think we must go a bit out of the box, and think further about the vality of the professors statement, and how it changes out of the circumstances. Initally the statement on the week before is inlogic, or invalid. But then the students realized it as such, and it became valid then.

Can you say that vality does change of time? Does vality change from the sight of the observer?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 31st, 2002, 5:20am

I don't quite understand what you said before... Can somebody smarter explain it to me?

on 07/31/02 at 01:03:22, anshil wrote:
 His argument became valid after he dealt out the test, so he didn't lie at the end.

The question I am having is: What does surprise mean? Does it mean surpise-all-the-way, or just "I won't tell you when the test is"? If it is surprise-all-the-way, he still lied... If it the other interpretation, the students were wrong.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 31st, 2002, 6:52am
It's surprise all the way, or otherwise he said something different as he ment, and we suppose that he said the thing he ment.

In the end his invalid statement is valid, since the students considered it to be invalid and didn't expect a test, so they got a surprise test in the next week as it was proclaimed.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by icon on Jul 31st, 2002, 9:17am
i wanted to give my thoughts on this issue:

i feel that this is the main problem here:

(am gonna comment my thoughts on each quote from the riddle which i think is easiest to follow)

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

(perhaps this is true but its ONLY can be true at 1 point of time(area thursday afterschool/friday befor school, which makes it before then a total suprise(however following this kind of logica we can take out friday).

Similarly, the quiz can't be on Thursday, because we know it won't be on Friday,(this is a mistake because you cant know that it CANT be on friday until thursday night! therefore in quiz will be given on thursday it will be within rules and a surpise for student(since this is the flaw in thier logic) and if the quiz doesn't happen by Wednesday, it'll be obvious it's on Thursday [because it can't be on Friday)]

(this is same as statement above, you will not know if the quit be on thursday until thursday evening and not by wednesday evening since it can be thursday or friday(mind you friday exception only comes into play in thursday evening!)).

Same thing for Wednesday, Tuesday and Monday. So it can't be on ANY day, so there's no quiz next week!"

its an intresting problem but i dont see how you can predict the days when it will happen since its a even 20% chance to do so

chance becomes much higher as each given  day goes by however you still cant really predict it

i think teacher 'confused' students by saying he will not give quit if they can predict, therefore giving students a false sense(that there is an answer if you look hard enough(logically) you can reach it) while there wasnt 1

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 31st, 2002, 12:13pm

on 07/30/02 at 21:50:15, Nicodemus wrote:
 Gamer, I think I see the distinction you're talking about. Let me see if I can restate it:We can use a stricter definition of surprise meaning that the students won't know when the test is, even the night before. This slight difference in meaning has major implications: If the professor is limited to picking a day that will be a strict surprise, then he cannot choose Friday because at some time before Friday it becomes predictable. Specifically, Thursday evening it becomes obvious the test is Friday.Thus we eliminate Friday completely. Our strict definition removes the temporal prerequisite from the proposition by disallowing days that will ever become predictable.My other argument hinged on the ability to discard the unconditional conclusion "it can't be Friday". Under the strict surprise definition, this is now a valid conclusion. Therefore we can follow the logical chain of the reasoning put forth by the students... Each successive day uses the same logical argument as the available days in the week diminish. Eventually, there are no days remaining, so there cannot be a test.Gamer, I hope I summarized your argument correctly? If so, we can at last explain the problem (in mind-numbing detail). The flaw in the students' reasoning (since evidently there was a quiz) was in their interpretation of the word "surprise". They followed the above chain of logic from that premise, while the professor followed the logic I and others detailed before.Now Mr. Wu can remove the "unsolved" mark on this problem! :)

Yes, that is *Exactly* what I have been saying. Sorry I missed this before!

I also think what Icon was saying, "i think teacher 'confused' students by saying he will not give quit if they can predict, therefore giving students a false sense(that there is an answer if you look hard enough(logically) you can reach it) while there wasnt 1 " is true too.

Summary solution: The students thought the teacher meant "strict surprise" or "surprise-all-the-way" but the teacher really just meant "You won't know what day I am going to test you"

I am not saying the teacher's definition precludes him from giving the test any day.

I am also not saying he favors days at the end of the week (like Friday and Thursday) more than days at the beginning of the week (like Monday and Tuesday) when it is Saturday or Sunday.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by icon on Jul 31st, 2002, 12:27pm
i think the question here is not how to predict what day teacher was to give quiz(since most people went this way to explain how it wont be a suprise on friday if by thursday its not given so it wont be given at all then) but what the flaw in students logic

flaw is: they tried to logically predict what day teacher will give the quiz, which cant be done unless you doing a guess
(all the logic about it cant be friday/etc is pure bolony since it doesnt work if he gives quit on monday then all the stuff about other days cant be quiz days wouldnt work)

this is a case of over thinking the problem hehe

perhaps william can comment

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Jul 31st, 2002, 12:32pm
I tend to agree with your analysis, Icon.

Anshil, you raised the question:

Quote:
 let's look on the professor. Can we really overlook that he flat out lied?

An interesting point. I discarded this option because I think that the riddle has a tendancy to collapse without it. If the professor lies, then we have no proposition on which to base any logic, since we must discard his statements. The answer to where the flaw in the students' logic was becomes "believing him". :)

But then you went with the assumption, as far as I can see, that the professor didn't lie in the sense that his original prediction came true: there was a quiz and it was a surprise.

Yet we can arrive at this conclusion without assuming his statements are false in any way. This leads back to the multiple interpretations of "surprise"; his original statements appear false if you choose one meaning (total surprise) yet appear true if you choose the other (not predictable).

Quote:
 In the end his invalid statement is valid, since the students considered it to be invalid and didn't expect a test, so they got a surprise test in the next week as it was proclaimed.

If I read your argument correctly, then he meant strict surprise, just as the students used in their logic. If the test was ever predictable before the moment it was handed out, it would be cancelled. Since the students reasoned from this logic, it was predictable and cancelled.

Yet he didn't cancel it (and that's a surprise). The cancelled/not cancelled paradox is brought about by the fact that we started with a false premise; specifically, the professor lied when he said he'd cancel the quiz if it were predictable. The paradox is simply the result of reasoning from false premises -- a proof by contradiction.

There is no contradiction between his other statement that the quiz will be a surprise. It was. But that seems to me irrelevant, since we've demonstrated that the puzzle's premise itself was false for that interpretation of "surprise".

Hence, also given the assumption that riddles contain only true premises (otherwise we can break rules willy nilly and they aren't riddles!), we conclude that the professor was interpreting the meaning of "surprise" differently, which was our solution.

How does that measure up to your thinking? Convincing or did I miss something? :)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 31st, 2002, 1:03pm

In my understanding a suprise test is a suprise test, it actually can't be announced. That's something different than a test on an untold day. A suprise test can't be on a day where it's 100% sure it will come. I think it's not right to say well the professor interpreted his own words wrong and actually ment the on any day variant.

The professor didn't lie in the part when he told the students the test will be canceled if the guess the day correctly, did the students tell him on which day the test will be? No. However this test will be canceled when guessed rightly is an enhancement to the puzzle by I.M._Smarter_Enyu who posted it here. It's actually not necessary to the puzzle soI don't think it's important to the solution.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Jul 31st, 2002, 3:18pm

Quote:

Yes, though I'm still open to further ideas. :)  But this is the difficulty with such subjective problems; what one person might consider a sufficient explanation isn't convincing to someone else. (And that's just human nature, really.)

After reading your (great) comments in the other thread on this puzzle, I wanted to expound a bit more on some of my comments to clarify my position.

Quote:
 In my understanding a suprise test is a suprise test, it actually can't be announced.

This is quite an interesting point and gets right to the heart of the matter.

As I see it, the entire puzzle revolves around how we interpret the word "surprise". I would agree that your statement is correct for the strict interpretation of surprise; once we know that there will be a test, it can no longer be a total surprise. (If you remove the guessing of days, this is basically what the students attempted to prove.)

However, I put forth that you can still have a surprise test in  a different sense; this is the loose interpretation of the word. The exact time of the test is a surprise, in that it's unpredictable, even though we know there will be a test sometime. I think that this is a valid reading of the professor's statements. (Heck, I even had a professor who gave announced-surprise tests like this!)

The entire problem hinges on this dual interpretation of the word "surprise". The apparent paradox in the puzzle comes from mixing these two different meanings in one context. The students consider it a strict surprise while the professor considers it a loose surprise.

Depending on how "surprise" is read, the problem changes. The students' logic is both fallacious and correct. The professor's proposition is both upheld and violated. Perhaps your comparison to quantum mechanics is quite apt?

Although this isn't a simple "solution" to the puzzle's question, I think that this answer serves to explain the puzzle itself. And that I consider a satisfactory resolution of the problem.

(Of course, it is always possible that this description is flawed. I'm not convinced --yet-- that it's inconsistent with any of the offered counterexamples, though.)

Quote:
 The professor didn't lie in the part when he told the students the test will be canceled if they guess the day correctly, did the students tell him on which day the test will be? No.

True. The students attempted to argue that any day would be impossible; their arguments only work for their strict interpretation of "surprise", though. From the professor's point of view, he has kept his word, as they haven't guessed the day. And if the professor is using the loose definition of "surprise", there is no way they can determine the day (prior to Thursday evening), other than blind guesswork.

Another reason that I think this explanation holds water is that we can quickly resolve the puzzle by forcing a single interpretation of "surprise". (Collapsing the quantum state, I guess the analogy would be.)

If we use the strict meaning of "surprise", then it must be true that the students cannot deduce the time of the quiz at any point before it is given, else it would be cancelled. This is consistent with the students' logic. Since the test happened regardless, the professor must have lied in saying he would cancel it (but not that it would be a surprise).

If we use the loose meaning of "surprise", then we can see the flaw in the students' reasoning. The time of the quiz cannot be deduced before Thursday, since there are two equally possible days. The professor's statements are consistent.

Forgive my long-windedness! I would like to concur with your conclusion that this problem is deceptively difficult.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by icon on Jul 31st, 2002, 3:33pm
hehe

the most intresting part of this is basically the arguments both sides brought on definition of a supprise

i had a teacher back some years ago who i totally loved9as you can love any teacher in that sense of the word) as a person he was the person who actually was very funny and sometimes broke some rules but same time was pretty strict yet give u space if you justify it :>

and once in a while he should use some uses of physics in real world, which would associate with a 14 year old mind so he would go home and actually be intrested in that and not to go out and try to do all but think about the school work

reminds me once a poster he had about throwing a curve ball how the spin and the way you hold the ball makes it curve by creative a some sort of pressure so when it reaches it, it collapses and curves down(now mind you back then i was all sports and not lot of physics but this made me actually spend few days researching, where i found few other grips which would make the fall curve not from 12/6 but from 1/7 even 2/8(imagine clock)

what all this has to do with this ... well simple:> perhaps the proffesor knew that there is no way you can predict the quiz before thursday night(duh) with the information he gives

chances are this class had something to do with sciences so maybe his goal was to get people intrested in skipping the quiz and therefore do the research to skip(reverse psychology)
without realising so and therefore probably studing for the quiz itself :)

now i can be 100miles off course but i think this is intresting side point :>

Title: Re: anyone know the answer to the pop quiz riddle?
Post by David Lischka on Jul 31st, 2002, 4:53pm
Any old puzzle but a good un'

I believe that the Professor cannot actually fulfil his promise, in other words his constraints are mutually contradictory.  Just like trying to divide 1 by 0 and get an integer answer.

I have never seen any better answer than this.  But just in case there is an answer, let me recast the problem as a card game.

Dealer has 7 cards exactly one of which is the Jack of Spades.  He lays the cards face down in a row in front of the player.  The player turns the cards over, in order, one by one until he feels sure that the next card is the Jack (if he turns the Jack over by mistake the round ends there and then).  He then bets whatever he likes that the card is a Jack and turns it over.  If he's right then he collects from the dealer, if wrong then he pays the dealer. (The player pays a penny per round and if he turns the jack before he bets then he loses just his original penny)

Clearly the dealer has to surprise the player by placing the Jack otherwise he loses big time.  So he can't afford to place the Jack in the 7th and final position in case the player turns over the first 6 and is then sure that the Jack is the 7th and final card.  So far the puzzle is the same as the pop quiz.

But

Play the game many times.  Would you rather be the dealer or the player?

Hint every time I play this I lose a lot at first and then end up winning \$100, 000, 000 !!

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gamer555 on Jul 31st, 2002, 5:50pm
I got it! Be the un-dealer, and bet lots of money, the first time, and double your bet each time. You will end up winning sometime, and win the same amount of money that you bet before (when you bet lots of money).

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Jul 31st, 2002, 11:43pm
I believe that the Professor cannot actually fulfil his promise, in other words his constraints are mutually contradictory.  Just like trying to divide 1 by 0 and get an integer answer.

Juhu, somebody agreeing with me.

Hmmm, can you thing about other examples with the same paradox, I have some classics, but they are a bit different, as you can just say these are invalid, in contrast of the professor ones.

"This sentence is false".

Sign on a bus station: "The last bus will not drive".

"The barber in a certain village is a man who shaves all and only those men in the village who do not shave themselves."

However these are not the kind of paradoxes I'm really interested in. I'm interested in other situations similar than the professors when a sentence cannot be valid or invalid at all. Like "There is a surprise quiz tomorrow". (If you judge the sentence as invalid, you're suprised tomorrow by the quiz, so it was actually valid.)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by aliks on Aug 1st, 2002, 10:32am
Like everyone has been saying the flaw in the students logic is that the students are assuming "surprise" means  "you will never be able to predict the day of the quiz" (which they prove the professor cannot do) as opposed to just "you can't say right now which day I will choose"

You can see the difference if you look at the card game I suggested above.  If the player has to predict in advance which card is the jack, he stands little chance even if he knows that the dealer is placing the jack at random among the first 6.

However, the dealer must have a strategy for placing the jack that withstands the player turning a single card over, or two cards or even 6 cards and this is what makes life hard for the dealer.  On top of this the game can be repeated so the dealer strategy may become obvious.

If you can persuade someone to play this game for money you should start off as the dealer and do something like a random choice amond the first 6 cards.  The player will likely come to the conclusion that he has only a 1 in 6 chance and since he pays a penny for each round, the odds are stacked against him.  Offer to let them become the dealer and now they are faced with the professors dilemma.  If you know for sure what their strategy is, then you can beat it.  For example if they place at random among the first 6 then each round you turn over the first 5 and sooner or later you will not have seen the jack in which case you bet heavily.

The problem actually goes deeper, because the dealer is giving away information about his strategy with every round.  The dealer has to stay ahead of the players attempts to guess his strategy, and its quite an interesting challenge to decide what they each should do.  I think there was a competition last year to write paper, scissors, rock programs to take on all challengers.  Some interesting ideas were on display (see Slashdot history for details)

(by the way if the player can't work out the dealer strategy then he tends to lose because he pays a penny a round to play and the odds are against any random choice he might make.  So doubling up on bets each time won't work)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by aliks on Aug 1st, 2002, 10:33am
Like everyone has been saying the flaw in the students logic is that the students are assuming "surprise" means  "you will never be able to predict the day of the quiz" (which they prove the professor cannot do) as opposed to just "you can't say right now which day I will choose"

You can see the difference if you look at the card game I suggested above.  If the player has to predict in advance which card is the jack, he stands little chance even if he knows that the dealer is placing the jack at random among the first 6.

However, the dealer must have a strategy for placing the jack that withstands the player turning a single card over, or two cards or even 6 cards and this is what makes life hard for the dealer.  On top of this the game can be repeated so the dealer strategy may become obvious.

If you can persuade someone to play this game for money you should start off as the dealer and do something like a random choice amond the first 6 cards.  The player will likely come to the conclusion that he has only a 1 in 6 chance and since he pays a penny for each round, the odds are stacked against him.  Offer to let them become the dealer and now they are faced with the professors dilemma.  If you know for sure what their strategy is, then you can beat it.  For example if they place at random among the first 6 then each round you turn over the first 5 and sooner or later you will not have seen the jack in which case you bet heavily.

The problem actually goes deeper, because the dealer is giving away information about his strategy with every round.  The dealer has to stay ahead of the players attempts to guess his strategy, and its quite an interesting challenge to decide what they each should do.  I think there was a competition last year to write paper, scissors, rock programs to take on all challengers.  Some interesting ideas were on display (see Slashdot history for details)

(by the way if the player can't work out the dealer strategy then he tends to lose because he pays a penny a round to play and the odds are against any random choice he might make.  So doubling up on bets each time won't work)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Nicodemus on Aug 1st, 2002, 12:35pm

Quote:
 I believe that the Professor cannot actually fulfil his promise, in other words his constraints are mutually contradictory.  Just like trying to divide 1 by 0 and get an integer answer. Juhu, somebody agreeing with me.

Actually, I think that I also agree with you, for one specific interpretation of "surprise". I think our opinions differ on whether that word can be used in different ways?

Quote:
 I'm interested in other situations similar than the professors when a sentence cannot be valid or invalid at all. Like "There is a surprise quiz tomorrow". (If you judge the sentence as invalid, you're suprised tomorrow by the quiz, so it was actually valid.)

Perhaps you'll think I take a simplistic view of things, but I group that sentence (along with "This sentence is false.") into the category of illogical sentences. It appears to be a logical sentence and tricks us into thinking of it as such. But, on closer examination, it becomes clear it is malformed and carries no meaning.

Here's my reasoning, in fancy-pants form for clarity :)

1. Assume "There is a surprise quiz tomorrow" is a logical sentence
2. A logical sentence is one that is true, false, or unproven (true xor false). [definition]
3. If we assume (1) as true, we get a contradiction [it is no longer a surprise, so it's description is contradictory]
4. If we assume (1) as false, we get a contradiction [we do not expect a quiz, and as it occurs, it is a surprise]
5. Thus (1) is cannot be true or false [from (3) and (4)]
6. Thus (1) is not a logical sentence [proof by contradiction of (2)].

(You might quibble with the definition in 2. That's my view and, hence, the results make sense from this basis.)

The sentence's ability to produce results that reverse it's interpretation fools us into following the logic round and round. What matters is that we simply find a contradiction under logical interpretations. It's still a sentence, just not one that we can analyze using logic; further work would be fruitless. Given that understanding, the paradox is gone; we know the sentence is illogical and we are done.

Does anyone else agree with this interpretation of such paradoxes? I'd be quite interested in hearing opinions. (I knew I should've taken more logic courses in college!)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Jonathan the Red on Aug 1st, 2002, 1:44pm
This problem is known as Newcomb's Paradox, or sometimes The Unexpected Hanging.

I maintain that the professor can give the test on Friday and still have it be a surprise. Here's how:

Thursday comes and goes. No test.

The students reason as follows: "The professor told us we wouldn't be able to figure out what day of the week the test is. But we know the test must be tomorrow, and as we all know, the professor never breaks his word. Therefore, he can't give the test tomorrow. We're not having a test at all."

The next day, to the great surprise of the students, the professor hands out the test.

This is where the logic of the student who, on Sunday, deduces that there will be no test at all that week, falls apart... at the very first step. As soon as you conclude that the test cannot be on a given day, you make it possible for the professor to give the test that day as a surprise.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by icon on Aug 1st, 2002, 2:15pm
nice! not much else to say here but agreement

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Alex on Aug 2nd, 2002, 9:59am
I think everybody's overcomplicating it with their definitions of "surprise" and terms like "surprise-all-the-way." The definition of surprise is that the students don't expect it, simply put. I think Jonathan has the right idea. It can be on any day, even Friday. We all seem to agree that Friday is the only day that can be logically "guessed" away, but assuming that Wednesday comes and no quiz has been given, the students will likely guess that the quiz is on Thursday (since it can't be on Friday right?). Since they have already guessed, they will be surprised when Friday comes around and they receive a test.

There is no solution to the riddle the professor has given the students. There are only guesses. The students' flaw in logic was to assume that the professor would give the quiz on the most unpredictable day, when in fact, all he needs to do is present the quiz on a day that the students don't expect it on.

It has nothing to do with the definition of "surprise."

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Jonathan the Red on Aug 2nd, 2002, 4:15pm
BTW, as I said in a thread on the CS forum: I'm a complete gimboid. This isn't Newcomb's Paradox. Newcomb's Paradox is something else.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Scott Schneider on Aug 6th, 2002, 3:58am
I think that the student's flaw is fairly simple:  their logical argument only works if it IS Thursday.  If today IS Thursday, and there has not been a quiz, then it must happen on Friday.  No ambiguity.

But if today is Wednesday, then there are two possibilities left:  Thursday and Friday.  Now there's ambiguity.  What was true for Thursday does not carry backwards to Wednesday.  Their implicit assumption is that it does.

What they're trying to do is extrapolate backwards, but that doesn't work because we must be on that day to eliminate possibilities.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 6th, 2002, 4:27am
I think the paradox here is additionally complicated around the core in many ways:

* first: here it is presented as a riddle, but in fact it's a very interesting phenomean, more than a "riddle".

* second: the number of days is set to 5, which are a lot. However the number of days don't matter, think about 3, about 2, or even about only 1 day. ("tomorrow will be a suprise quiz").

* third: This all happens in the school system, we're all still hurt from our youth and our own school time, with the god given paradigm: the professor can't be wrong (or you'll only loose big if you dare to think otherwise). Think about how things would be like the origin of this paradox. For example "there will be an unexpected fire alarm training next week", (which can't be on friday, since we would expect, then on thursday ... ) or honoring point two. "There will be an unexpepected fire alaram training tomorrow or the day after tomorrow". (Can it be on the day after tomorrow, and tomorrow then?)

* forth: The riddle asks "what have the studends done wrong?", is this a valid question after all? Have they done anything wrong? This must not be the point of this phenomenoen.

* five: suprise is actually not too well defined here, just suppose what would be if it means on any day (thats a trivial case), or surpise-all-the-way (fire alarm training), thats the interesting case. Just because there is a trivial case also doesn't mean we should ignore the other, more interesting case. (just alter the riddle in mind when the professor would have said explicitly, surprise-all-the-way)

- Of what other timelines can you think this situation could have gone through?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Ryan Lawrence on Aug 6th, 2002, 11:40pm
Okay, I want to preface this by apologizing if any of this has been said already, I am short on time at the moment and have not read all posts.

However, from what I have read so far I think you guys are on the wrong track. I do not think that any of it relies on the idea of being a surprise or somesuch idea. Seeing as this is a LOGIC class, let's use logic terms and break it down syllogistically:

A: The test cannot be Friday because if it doesn't happen by Thursday, we know it will be Friday.

B: The quiz cannot be Thursday because of Premise A, so if the Quiz doesn't happen by Wednesday, we know it is Thursday.

C: Apply the same theory to Monday and Tuesday.

Conclusion: There is no quiz.

The glaring error to me here is that Premise B and C are purely based on Premise A. However, Premise A relies upon some strange time travel that allows the students to see the future and know that no test will occur until Thursday.

Basically, all of the logic stems from the fact that there is no test Thursday, but they do not know that there is no test Thursday UNTIL THURSDAY COMES ALONG; therefore, the test can come "unexpected" Monday, Tuesday, or Wednesday because until Thursday their primary premise is not true, and if one premise is untrue, the syllogism is false.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Steven Noble on Aug 8th, 2002, 4:19am
First off that sort of reverse recursion is a completely acceptable form of deduction.  As a popular example think of a finitely iterated 2 player prisoner's dilemma game.  Both players will defect every iteration.  However the proof for this  is found by the exact same form of "strange time travel" that you found erroneous.  This is a well accepted proof and not wrong.  (upon re-reading my comment I feel it is important to note that there is not a causality link from "well accepted" to "not wrong."  In this case both happen to be true.)

So the question is what did the students do wrong.  What they did wrong was tell the professor of their conclusion.  If they kept their mouths shut, when they received the test they could say "nope... we knew it was going to be today" and then give their explanation as to why.
Here would be their proof as to why.
By the earlier proof we know there is no surprise quiz.
So the assertion "There is no surprise quiz or the quiz is on Tuesday"
is true. (Let us think us this as ~Q||T)
Then we have the assertion "There is a surprise quiz" (or Just Q)
Which leaves us with the following
(~Q||T) & Q => T

Even better every day at 1 minute to midnight a student could call the professor and say he guesses the test will be tomorrow.  There doesn't seem to be a limit on guesses.

Note there is a time when the latter would be a bad idea.  If the class were on a bus heading west.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by horseisahorse on Aug 8th, 2002, 9:15am
Going back to the card game with the Jack of Spades, I think that a dealer's optimal strategy would occasionally allow for making the 7th card the Jack, since you might elicit a large bet after the turning of the 5th card.

I think that some game theory is called for here.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 8th, 2002, 9:40am
Steven can you tell us about the 2 prisoners?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gerard on Aug 8th, 2002, 1:47pm
Hello all.  This has been an interesting riddle, and I have enjoyed reading the discussion.  To put in my two cents and my theory…   I want to tackle just the main points (no dawdling) and set up some parameters to work from.  I will propose my answer to the teacher’s question with a short proof, and then state why the students were wrong in their logic.

What is a ‘surprise?’  I propose it is quantifiable, you can be mildly surprised, fairly surprised, or totally surprised.  For the first part of my response we will correlate surprise with unpredictability. To give me subject matter to work with, I will represent our predictability and surprise with a percentage, where 0% is totally unpredictable (totally surprised) and 100% is predictable with assurance (not a surprise, expected).

BTW, this is all based on the assumption the test will happen.  If the professor says it will happen, I (as a student) have to assume it is valid and true.  If I guess the correct answer and the professor smiles and says “Correct” and because of the correct answer there will be no surprise test next week, then all statements were valid and true.  To guess incorrectly and without a professor’s statement to cancel the test, all statements would still remain valid and true.

So let’s go through the week backwards.  If it is Thursday night, it would be 100% predicted to get a test on Friday (we would fully expect it, and it would not be a surprise).

If it were Wednesday night, at first glance one could say there was a 50/50 chance between getting the test on Thursday and Friday.  But with the 100% predicted surprise found the night before a test on Friday, and assuming the goal of the teacher to be to maximize the surprise, we would expect Thursday to be the more likely candidate between the two.  (Now the numbers I am going to throw out are not perfect, but meant to generally illustrate the growing surprise for the diminishing guarantee of predicting the day of the test.)   I would say there would be 80% predicted to get the test on Thursday (we would mostly expect it but the remaining Friday would still be a possibility).

(just a side note, since ‘surprise’ is a subjective word, the test could still be on Friday.  It would just be a lame surprise)

If it were Tuesday night, at first glance we could see a 33/33/33 chance between the three remaining days.  We will take the previously calculated numbers with the three possibilities, with 100% predicted if it’s on Friday, and 80% predicted if it’s on Thursday.  We would use that weighting of chances of predictability to guestimate there would be 60% predicted if a quiz happened on Wednesday.

What I’m getting at is the more days you have ahead of you before the end of the week (Thursday night’s 100% predicted Friday test) the more days to be possible candidates, therefore increasing the unpredictability from that day, and therefore increasing our present definition of surprise.  So the most unpredictable (hence largest surprise) day would be a Monday test with this logic.  On Sunday night a student would have all possible days ahead of them for the test to be on.

The web page said the test happened on Tuesday, this makes me think twisted thoughts.  I will try to explain.  If the goal of the logic professor is to maximize surprise, he would choose a day the student’s least believe the test will be on.  However, he asked them to make a guess with the guarantee of no quiz if they guess correctly.  The difficulty for me in this question is the lack of absolutes.  Surprise is subjective and conditional.  The professor asks a question, using logic for students to figure a test day based on the amount of ‘surprise’ it would cause.  If there was an obvious linear logic to give you a day that is least likely and therefore most surprising, all the professor would have to do is choose a day other than the logical answer and that too would be a great surprise.  It’s a paradox in a paradox when the variables are considered absolutes.  So I hope my guestimations aren’t too confusing or too non-academic.

I am pretty solid on my answer and believe it is the best possible answer to the professor’s question.  The way to approach this question is to see a growing unpredictability on the night before class as the number of days that were ahead of us to the end of the week went up.

With the above in mind.  The student’s flaw in logic was their definition of a surprise.  They assumed it had to be a total surprise of unpredictability that hit when the test was passed out.  They accepted no other possibilities, such as a mild surprise when there were just a few days left before the end of the week.  When we are dealing with subjective items like surprise, we have to take a more weighted and less yes/no approach.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 8th, 2002, 10:06pm
Okay Gerad, for you I modify the professors statement, to your defintion of suprise.

"There will be a test next week which has not a 100% predicted suprise".

What now?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gerard on Aug 9th, 2002, 12:24pm
Hello all,

I would take the new definition of a surprise, use the logic I mentioned above of the increasing unpredictability given the number of possible days in front of you, and say the quiz should have been on Monday.  But of course, the riddle states the quiz happened on Tuesday.  So I don’t know the answer to this one, and to be honest have to question if there is an unequivocal way to determine a day given the limited info posed in the riddle.

Besides that, if there IS a straight logic that everyone would agree with to determine the day.  A professor could take that expected result, give the quiz on a different day, and the general ‘surprise’ would still happen and the professor wouldn’t be wrong.

Quite the difficult riddle.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by ilia Denotkine on Aug 9th, 2002, 1:39pm
The quiz can't be on friday, because the students will know it on thursday.
they can't say surely on wednesday when they will have the test because it can be and on thursday and on friday. Use the sane logic for other days.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Ryan Lawrence on Aug 9th, 2002, 4:35pm
The discussion here and the basis for using the recursive logic is that the surprise quiz has to actually be a surprise in the literal sense of the word. This is a Catch-22 because if there were any way for the students to figure out when the test was, it would no longer be a surprise.

What I am failing to comprehend is why we are overanalyzing the term "surprise quiz." As I see it, the point of calling it a pop quiz is just so that we know it is not set on a single day, and can be on any day that the professor chooses. If we take the riddle at more of a face-value the answer I provided earlier makes sense, because the recursive logic relies on the premise of overanalyzing "surprise."

Title: Re: anyone know the answer to the pop quiz riddle?
Post by bling0 on Aug 10th, 2002, 1:14am
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.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by George Wright on Aug 10th, 2002, 6:07pm
There are two versions of this forum, I just wanted
to transfer my thoughts to what seems to be being read
most recently

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.

The students decided to try to make the axiom set consistent
by ignoring Axiom 2. While the professor gleefuly could give
them a quiz, because there was no way they could prove
anything useful from a contradictory set of axioms.

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.

Re-reading the problem as stated however, I just
noticed that by saying he will cancel the quiz
if they prove him wrong, he's given himself an out.

He could conceivably hold the quiz on Friday, and
then cancel it when they prove it to him, with this
option he's OK and can hold the quiz whenever.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Steven Noble on Aug 10th, 2002, 7:22pm
I've just been reading some of the recent posts and after reading George Wright's post I think I know the answer.  It was the fact that George listed the 3 assertions as Axioms.  It occured to me then that I was too treating those 3 assertions as axioms as well.  Of course they are not and that's where the students went wrong as well.  An axiom is something that can't be proven but is known to be true.  The days of the week thing can be proven since the def'n of a week is seven days so by definition the assertion is true.  And as for the professor's statment... well that appears to simply be an assumption and not an axiom.  I may tend to believe it is true but I don't know it.  So what does all this mean.   It means I came to the following realization.

Through out the course of the students proof they come to a contradiction: "there is no surprise quiz next week" and "there is a surprise quiz next week."  Well in a proof by contradiction, when one comes to a contradiction, any contradiction, he simply know that one of the original assumptions is false.  In this case there is only one option as to which original assumption is false - "there is a surprise quiz next week."  When we know an assertion to be false we then know its inverse is true.  BUT is the inverse of "there is a surprise quiz next week" "there is no surprise quize next week."  I would submit it is not.  First off we are talking about the future.   Since we are not omnicient there are always several possible futures.  So the assumption can be rewritten formally "For all futures there is a surprise quiz next week."  For those who have studied formal logic alarm bells should be going right now.  Becuase the inverse of any "For all... etc" is always an "There exists a... etc."  In this case the proper inverse is "There exists a future such that there is no surprise quiz."  Translated back in regular speak we get "There might be a surprise quiz next week."

The students logic is correct.  They simply drew the wrong conclustion.    It is equivilant to a proof that begins with "assume x=2" and ends with "x=3."  The correct conclusion drawn would be "x!=2" but these students might come to the conclusion "x=3."

---
ps  anshil I saw your message about the two prisoners and I promise soon I will explain who they are and what their dillema is.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 15th, 2002, 11:07pm
From what I can see, the solution to the question posed is much simpler than anything to do with the semantics of what constitues a "surprise".
The question asked is "what is the flaw in the students' thinking?". The answer to this question seems too simple to me, so I must be missing something, but here goes.

1) Assume no test by end of thursday (ok)
2) Then test must be on friday (ok)
3) So test cannot be on friday and still be a surprise (ok)
4) Assume no test by end of wednesday (ok)
5) by 3, then test must be on thursday (contradiction!)

The contradiction is simple. (1) assumed there was no test by end of thursday, but by (5) the test must be on thursday. Clearly, the logic used cannot be recursively applied in this instance, as it results in a self-contradiction.

In other words, the reasoning is valid on the first pass: if no test by end of thursday, then test must be on friday, therefore predictable. Nothing wrong with that. But you cant use the assumption of no test by end of thursday to then eventually (in the next recurse) show that it must be on thursday.

edit: this reminds me of a bugs bunny (iirc) cartoon where bugs keeps building a staircase higher and higher by taking steps from the bottom.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 15th, 2002, 11:52pm
Archon have you read all of this thread? or just replied at the end of it. You can't just disable the logic just because it results in a contraindiction, it's falid still and the core of all the puzzle. please read all the messages, there are really good ideas from some of the people in there.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 16th, 2002, 1:05am
I have read the whole thread, which is why I mentioned that I was surprised by the commentary on the semantics of "surprise".

I was not trying to say that the logic results in a contradiction and is therefore invalid, certainly you can have a logical argument resulting in a contradiction.
The problem is that the recursive application of the logic results in a condradiction with the initial assumption on which the logic is based. The recursive application still relies on the initial assumption of "no test by end of thursday" but then continuing the logic places it on thursday. Can't do that. You can see a similar thing happening in the Universal Truth Machine puzzle, where people are assuming that it works in order to prove that it doesn't.
I think things have gotten far too abstract on the path of what a "surprise" is, and has missed this fundamental point, which is the answer to the question posed.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 16th, 2002, 7:24am
I say recursivness has not much to do with the paradoxon, see the "there is a surprise test tomorrow" variation, there is no recursivness in there, and still holds true for all other traits of the puzzle.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by George Wright on Aug 16th, 2002, 7:19pm
Archon,

Although its a tad confusing given the way some people
have written the students proof I don't think that's at fault.
Here is another way of writing it that might make it clearer.

Since from the professors statement we assume there exists at least one quiz, on Monday through Friday

Either 1) the only quiz is on Friday,
or
2)there exists a Quiz on Monday through Thursday.
these are the only 2 possibilities

If case (1) were true the students would come in Friday morning knowing that there had been no previous Quiz.
And therfore be able to deduce that the quiz had to be on Friday. So under the assumptions that case 1) is true we
reach a contradiction.  Therefor case 2 must be true.

Now we know that there exists a Quiz on Monday through Thursday.  There are 2 cases

Either 1) the only quiz is on Thursday,
or
2)there exists a Quiz on Monday through Wednesday.
these are the only 2 possibilities

We repeat this procedure until we get to

There exists a Quiz on Monday.

Now there are no two possibilities, the only possibility
is itself a contridiction, and we are left to conclude that
the axioms that were used to formulate the problem
were inconsistent.

-George

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 16th, 2002, 8:20pm

Quote:
 Now there are no two possibilities, the only possibility is itself a contridiction, and we are left to conclude that the axioms that were used to formulate the problem were inconsistent.

Which axioms are those?
As far as I can see, there is nothing wrong with saying "assume no quiz by thursday, then quiz is predictably on friday".
And there's certainly nothing wrong with "there will be a quiz next week".

Quote:
 Either 1) the only quiz is on Friday, or  2)there exists a Quiz on Monday through Thursday. these are the only 2 possibilities  If case (1) were true the students would come in Friday morning knowing that there had been no previous Quiz. (...) Now we know that there exists a Quiz on Monday through Thursday.

No we don't. We were only able to eliminate Friday by assuming that there had been no test mon-thu. We cannot now "know" that the test must be on mon-thu, having just assumed that it wouldn't be in order to make our case.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Ryan Lawrence on Aug 17th, 2002, 6:46pm
George, the logic used still relies on, as Archon said, the semantics of a "surprise." The logic being used to discount Friday as being the quiz day is as such:

If the quiz does not happen by Thursday, the students would know it was Friday and as a result it would no longer be a surprise, therefore the surprise quiz cannot be on Friday.

Personally, that is not too convincing to me, especially since it relies on a *huge* assumption that the author of the riddle intended for there to be an inherent catch-22 (that you can stop the quiz if you find out what day it is on, but you can't find out when the quiz would be or else it wouldn't be a surprise).

"Sometimes a cigar is just a cigar," and in this case sometimes a pop quiz just means that the students aren't told what day the quiz is on. Therefore, to me the obvious answer is that the entire logical chain is based on the quiz not occuring on Monday through Thursday, so when the quiz occured on Tuesday, the students were surprised.

Maybe there are toe answers, the simple one and the long, drawn-out, complex answer that you folks are tryin to get to, but personally, I am quite content with the obvious.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by George Wright on Aug 17th, 2002, 9:49pm
Archon,
I should probably just let this drop, but I have far too much of a stuborn streak.

The 2 cases I presented in my argument are mutually exclusive, and cover all possible conditions.  If you
disagree with this statement, then please either show me
how they can both be true at the same time, or show
me how they can both be false, given that there exists
a test.

Next I chose to take each case one at a time.

Since case 1 and 2 were mutally exclusive, when I consider
the possibility that case 1 is true, I am allowed to assume that case 2 is false.  But only for this portion of the argument.
What I found, was that by assuming that case 1 was true I reached a contradiction.

According the the laws of sentential logic, If I know
that one of A or B is true, and then I find that assuming A leads to a contradiction, then B must be true.

There for when I show that not having a test Monday
claim that there must be a test Monday through Thursday

Also the Axioms I was refering to , were those outlined
in my message on this topic dated August 10th.

Cheers,

George

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 18th, 2002, 1:48am
OK, assuming your use of "sentiential logic" is correct, then the eventual conclusion is that there is no test, which is obviously false, and therefore clearly *something* is wrong. Agreed?

The problem is that each case is a self-contained unit. You cant start with one assumption in one case, and then rely on the assumption in a completely different case in order to complete the logic:

Case 1
Assumption: No test before Fri
Conclusion: Test must be on Fri.
OK.

Case 2
Assumption: No test before Thu
Conclusion: Test must be on Thu.

This is clearly NOT OK. The correct conclusion should obviously be "Test must be on Thu OR Fri."
We were only able to eliminate Fri in case 1. This is not case 1. You must IGNORE case 1 when you start case 2.

Continuing, the correct conclusions are obvious...

Assume no test before Thu. Then test must be on Thu OR Fri. OK.

Assume no test before Wed. Then test can be on Wed, Thu, or Fri. OK

etc.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 18th, 2002, 3:28am
Archon:
No we know on wensday already that there can never be a suprise test on Friday.

Again think about the "there is a surprise test tomorrow or the day after tomorrow" version. This still has the same problematic, and this thinking does definitly not work here.

Ryan Lawrance:
Of course as I've written above there are two variants you can look this upon, the easy one just meaing surprise by any time, and the interesting one meaning suprise-all-the-way.  Of course from practical view it isn't too interesting going after case 2, however from a scientific view it is a very interesting "gedankenexperiment" in the science of logic, you can't just render this case invalid just because you can find another view to look upon this.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 18th, 2002, 7:39am

on 08/18/02 at 03:28:44, anshil wrote:
 Archon:No we know on wensday already that there can never be a suprise test on Friday.

That doesn't matter.
As soon as a "surprise" (read, time unpredictable) quiz is announced for next week, we know it can't be on Friday and also remain unpredictable, because that would make it predictable *by the end of thu*. As I have said twice now, there is nothing wrong with this part of the logic. It doesn't mean the quiz can't be on Fri, it simply means that we will KNOW it is on Fri *by end of Thu*.

That means that yes, on Wed, we still know that there cannot be an unpredictable quiz on Fri. But we still do not know whether it will be on Thu or Fri. If Thu goes by without it, THEN we know it's on Fri, and is predictable. But we do NOT know on Wed that the quiz is on Fri, only that IF it is on Fri, we will be able to predict it at the end of Thu.

Let me put the riddle to you another way. I am thinking of a number between 1 and 5. I am going to count from 1 to 5. If you can tell me which number I am thinking of before I say it, you win, otherwise I win.
Are you going to tell me that I cannot actually be thinking of a number between 1 and 5? (analogous to the reasoning that results in: "there cannot be a surprise quiz next week").

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 18th, 2002, 9:21am
<i>Let me put the riddle to you another way. I am thinking of a number between 1 and 5. I am going to count from 1 to 5. If you can tell me which number I am thinking of before I say it, you win, otherwise I win.
Are you going to tell me that I cannot actually be thinking of a number between 1 and 5? (analogous to the reasoning that results in: "there cannot be a surprise quiz next week").</i>

Thas there is there cannot be a _mmm_ quiz next week, without surprise all the way. The analogous would be you think of a number, count from 1 to 5, and you tell me additionally , it's a number I can never predict (thats the surprise fact). This argument has something wrong in itself. Since it may never be a 5, since it would be a number I can predict at four, rendering your arument false and you would have lied. So if we're at 3 I know it may not be a 5, since you said it's a number I cannot predict, so it must be a 4, (or you would lie), but then again I can predict a 4 again, so it can't be that either, or you would have lied, about the predict (=surprise) part. It can also not a 3, 2 or 1 then. Got it now?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 18th, 2002, 3:32pm
LOL, it's not that I don't "get it". I am pointing out the flaw, which is the object of the riddle. You have done a commendable job of explaining the logic used by the students, but it's still flawed ;D

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 19th, 2002, 12:40am
I think we're at the point this discussion with you doesn't bring any fruits, no the logic of the students is not flawed, as pointed out already several times by several authors, the base of problem itself is flawed. And still it seemed you don't get what it means that the quiz itself disables itself for not beeing a surprise quiz.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Archon on Aug 19th, 2002, 4:45am
<shrug>

Quote:
 the quiz itself disables itself for not beeing a surprise quiz.

Quote:
 On Tuesday, the professor gives the quiz, totally unexpected!

QED.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by anshil on Aug 19th, 2002, 5:06am
He could also give the quiz on friday totally unexpected! (Since the studends thought there would be none)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Onetus on Aug 19th, 2002, 6:10am
Okay - I'm just going to concentrate on what I believe is the nature of the puzzle - why is the student's logic wrong - not what day the test is, or what the definition of surprise is. I would expect that these questions should be solved without arguing the definition of the question.  :-/

Oh and one more thing - I'm not even looking at whether the student's logic is even the correct way to guess the day - I'm solely interested in why their answer was wrong.

The students identify the following pattern. On day n their cannot be a test on a day > n.
Consider this - Numbering Monday to Friday 1..5
On day 4, they identify that 5 cannot be the quiz day.
On day 3, they identify that 4 or 5 cannot be the quiz day.
On day 2, they identify that 3,4 or 5 cannot be the quiz day.
On day 1, they identify that 2,3,4 or 5 cannot be the quiz day.
On day 0, they ... wtf? There is no day 0!

According to their logic if there wasn't a test on the current day then it can't be any day in the future. Which is great until this falls down for Monday. It's the first day of the school week - and according to their logic - they can't prove there will not be a quiz for this day.

So I humbly believe, that the flaw in the student's logic was that they took their "sequence" beyond the bounds of what they could prove and that was their mistake in the logic.  ;D

Perhaps I've missed something fundamental or maybe I was hit by the obvious brick, but hey, bring on the lions...  ;)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Florian Pflug on Aug 22nd, 2002, 2:00am
Hi

IMHO the flaw in the students logic is to assume that THERE WILL BE A TEST UNDER ANY CIRCUMSTANCES.

But this isn't what the teacher stated. His statement was "There wil EITHER be a Test, OR (if you would know when it takes place in advance) there won't be one".

On thursday night, the students know for sure that there won't be a test (unless it has taken place already).

But on wednesday night, there are two possibilities now
1) The Test is on thursday
2) There won't be a test.

Without knowing for sure that there WILL be a test, the students can't rule out (2). This qualifies as being "surprise" I guess.

Greetings, Florian Pflug

Title: Re: anyone know the answer to the pop quiz riddle?
Post by fingasj on Aug 22nd, 2002, 8:16am
I think the solution to this problem is in how the students predict the day of the test. In order to predict the test, they have to TELL THE PROFESSOR what day the test will be on. And they have to do this before the test is handed out.

So, if it were Friday morning, and there hadn't been a test yet, then they could say "The test will be today!" thus cancelling the test. Of course this statement would be false ;) but that's beside the point.

However, on Thursday morning, they can't say "The test will be today", because that would preclude them from guessing on Friday. There's no way the professor would let them guess more than once! And by saying that, they would make it possible to put the test on Friday.

So, every day they can either: predict the test, or say nothing. But (except for Friday) either choice could be wrong. Therefore, they have a chance to predict the test, but it's like the Jack of Spades game now, and they must play against the teacher. It's possible that they'll guess right, but more likely that they won't.

Assuming they're not going to guess until they're SURE of the test date, then the professor can schedule the test on any other day than Friday. I'm sure it would still surprise some people!

And convincing themselves that there cannot be a test means that they'll never predict the test, guaranteeing that there is a test, and guaranteeing that it's a surprise.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Shadow Wrought on Aug 22nd, 2002, 1:34pm
I just read this and have looked over the thread, and there are couple folks with whom I agree, but I think for the most part that everyone is not answering the question.

The question is "What's the flaw in the students' thinking?"

The flaw is that the question of whether or not the test happens on Friday is mooted if the test occurs prior to Friday.

In other words, the flaw was the unstated assumption that the ONLY reason why the test could not happen on Friday was that it would be obvious.  The test could ALSO be prevented from happening on Friday if it occurs earlier in the week.

The difference of course is that the the test cannot occur other than the day on which it exists.  A simple tautoligism, but I think that it is the basic flaw in the students' otherwise vigorous thinking.

My \$0.02 :D

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Mark Wusinich on Aug 22nd, 2002, 7:12pm
I do not have time to read every responce. But here is my explanation. The QUIZ can be on FRIDAY.  Because if it is Thursday night and the quiz has not yet happened AND they have not yet guessed then they will know it is the next day and can guess that it is on FRIDAY. If however they have already used their guess on any other day then the quiz will still be on FRIDAY. If they get unlimited guesses then each day they should guess that it is on that day.

Mark

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Patrick on Aug 23rd, 2002, 6:24am
Maybe I am being silly in my reasoning but what if the professor was to have a test on every single day. Perhaps the reason that he surprised them on Tuesday was that they had already had a test the day before. Nowhere in the question does it state that there is only one test.

This could also mean that the students predict that the test is on Monday. The surprise test for that day is cancelled since it can no longer be a surprise having been predicted, however since the students are sitting back content that they have predicted correctly they are taken by surprise by the quiz the next day. The flaw in the students thinking is that there is only one test.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Steven Damer on Aug 25th, 2002, 9:36am
My thoughts on this problem. (Warning: long)
The professors initial statement is:
There will be a quiz on some day next week, and it will be a surprise.
Which can be restated as:
There will be a quiz next week, and you will not be able to deduce the date of the quiz from this statement.
Consider the following similar statement:
The number I am thinking of is 5, and you are unable to deduce what number I am thinking of from this statement.
Interestingly enough, even if the number I am thinking of is in fact 5, simply adding the self-referential addenda makes you unable to deduce that the number I am thinking of is 5.  Consider - if the statement is false, you are unable to deduce anything from it.  If the statement is true, then one of the consequences of the statement is that you are unable to deduce the number I am thinking of from it.
Looking at the surprise quiz, the reasoning is similar.  If the professor's statement is true, you cannot use it to deduce the day of the quiz, and if it is false, then you cannot use it to deduce the day of the quiz.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gerard on Aug 27th, 2002, 8:39am
Hello all,

The riddle asks “what's the flaw in the students' thinking?”  However, I do not see a way to show a convincing flaw in the students thinking without proposing a sounder logic that the students could have used. Logic can be shown flawed if the subject matter were more black and white, but that is not the case here.  The thread has discussed what the professor means by ‘surprise’ and the varying importance of the professor’s willingness to cancel the quiz.

Simply being a critic will not conclusively answer this riddle.  We can pick at how the students answer differed from what actually happened, but I for one will not be convinced until I see a riddle answer that contains a more believable answer to the professors’ challenge.

(Felt the need to defend why some of the others and I keep discussing this surprise and the correct answer that the professor would have canceled the test for)

Given the modified not 100% surprise (thanks anshil  :) ) I still stand by my Monday answer and belief that the students mistake was not in logic but in the definition of the word ‘surprise.’

Gerard

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Marc Lepage on Aug 27th, 2002, 10:53am
Apply logic not to the student's reasoning, but to the professor's statements.

Consider these facts:

A There will be a surprise quiz next week.
B I'm telling you what day.
C You can figure out what day.
D I will cancel the quiz.

Then the professor said this:

A
!B
C->D

We also know from the meaning of the sentences themselves that:

A->!B
D->!A

So altogether we have:

A
!B
C->D
A->!B
D->!A

Since C->D->!A which contradicts A, we know !C.

The students will not be able to figure out which day. The test will be given. It will be a surprise.

The student's problem is assuming the truth of C; that is, that they can figure out what day the quiz will be on.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by ron barry on Sep 5th, 2002, 3:15pm
Wonderful analysis, Marc Lepage.  Now for the real question - would the prof have cancelled the quiz if they'd figured out that he was lying?  i.e., !C?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by James Fingas on Sep 6th, 2002, 6:45am
The prof didn't actually lie about C, he said "if C, then D", or equivalently, "D or !C". Obviously from the story, !D. The professors overall statement was still true, however, because !C.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by MikePeuser on Oct 6th, 2002, 11:20am
Lets apply formal logic to this problem:
A = write test
B = is surprise
The students conclude:

A  and B  ->  not B,
thus not A
This however is basically wrong!

Because A -> B  is identical to not A or B, we can re-write the students conclusion
not (A and B)  or not B
which is
not A or not B or not B
or just
not (A and  B)
which means: the test can be no surprise!
This however is all that can logically be derived. That means the teacher has "lied", or has stated something contradictical.

Setting themselves in a state of "surprise" is by no means backed up by "logic". But because they do it the teacher is proved right in the end.

Lets fill the logical structure with another content:
"You are very clever but you will not pass the exam".
Student might derive from the fact of beeing clever that he/she will pass the exam. (He/She might as well have argued the other way round...) Becaus of his/her behavior he/she fails the exam and the teacher is proved right.
No logic problem in it......

Mike

Title: Re: anyone know the answer to the pop quiz riddle?
Post by wingyen LAU on Nov 26th, 2002, 12:38pm
Hi ALL;
The problem with this riddle is the logic is in time.
If the teacher lets the students make a guess everyday, then it can't be on Friday, since when u come in on Friday, u either had the test or u didn't.  If u didn't that means, coming in on Friday u know u'll have a time, and since u can say to the teacher u know it's today(friday) so it's not a surprise.  But M-TH can't be ruled out.  BUT one funny thing about this one is that the forward process works.  If the students can guess everyday then the students can get away by saying they know there's a test on Monday, so it can't be on M.  Then they say on tuesday that they know it's on tuesday and on tuesday the teache can't give a test, and the students do this till friday. =)

If the teacher lets the student make one guess on Monday ONLY, then it could be on any day.

Wing

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Kozo Morimoto on Nov 26th, 2002, 11:49pm
Has the comments on this site:
http://jimvb.home.mindspring.com/pobkuiz.htm

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Gowtham Goli on Nov 27th, 2002, 9:50pm
Finally every single day the students are expecting the prof to give a quiz and the prof never gives the quiz.
So as long as everyone studies really hard everyday the prof doesn't give the quiz, he cannot due to his statement, unless the students don't study wherein he can give his quiz because they are not expecting it.
solved?
:D

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Dylan parrish on Nov 28th, 2002, 7:32pm
You are all over looking the fact that he can be totally random.  The students can have all there damn logic.  If they told him that they didnt think there was a test at all during the week, then whats to stop him from giving one on tusday.  Logic cannot account for randomness.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Joe Pellino on Dec 12th, 2002, 3:06pm
Ok I think you are stupid if you haven't thought of this:  The test isn't planned for a specific day.  The teacher can change the day if he wants to.  If the kids say it is on Tuesday, he can change it to any other day.  This will make the kids think, which is what any good teacher wants.  ::)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by fenomas on Dec 15th, 2002, 9:46am
Seems fairly simple to me...  People seem to assume that the professor will set the quiz attempting to prevent the students from figuring out when the quiz will be, but the puzzle says nothing to contradict the idea that he may have chosen a day at random. He doesn't stipulate whether the students must (or can) use logic to figure out the date, or even if figuring out the date of the quiz is possible! Since the problem doesn't stipulate those things either, the students may not rely on them. Thus, they cannot assume that the professor would not put a quiz on Friday since people would realize the date after Thursday's class.

In general, the puzzle is simply too vague-- it gives no information about whether the students must guess before the week of the quiz, or if they can guess anytime, or how many times they may guess, or if a correct guess must be accompanied by a logical explanation of why it is correct. It's just a vague puzzle-- that's why it doesn't have a pat, satisfying answer.

If there's a problem with the above, please let me know....

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Joe Pellino on Jan 9th, 2003, 6:16pm
Thank you for agreeing.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Jiterbug on Jan 10th, 2003, 2:33am
A given fact in this riddle is that there will be a test next week!
If it is Thursday night of course the test will be on Friday, but how did the students manage to refrain from guessing a day until Thursday night?

The deductive reasoning behind Wednsday, that also set a precedence for the other days, states that the test MUST BE on Thursday, as otherwise it would be on Friday which is no surprise. The problem with this statement is that Friday is not a surprise if and only if it is Thursday evening! Since the students must guess prior to the test they have to guess on Wednsday night if they believe that Thursday would be the day. However if they are forced to submit a guess on that evening then Friday has just as much likelihood of happening as does Thursday.

Another way of looking at this problem is to consider a hat with the #s 1,2,3,4,5 in it (each number represents a day in the week), and an envelope with one of these numbers written on it (as the day is a surprise, you can consider the number as random). You can choose to do one of 2 things:
1) Guess up front which number is in the envelope, in which case the envelope is openend and your guess is verified.
Or
2) Let the numbers be drawn from the hat, and at any point in time you can stop the drawer of numbers and submit your guess (provided that the number in the envelope hasn't been drawn yet, if it would have been drawn, then the drawer would stop, as they themselves know the number, and prove to you that it was drawn by opening the envelope).

Would you deduce that there is no number in the envelope?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jan 10th, 2003, 3:36pm
Okay, consider the following variation:

The teacher announces that he will give a test next week, but the students will not be able to determine which day it will be until he walks in with the test. In deed, if before the test occurs they can show him with solid logic that his announcement is false  (either by determining which day the test is on, or by showing that he cannot give the test under those conditions), the test will be canceled and everyone will get 100%.

The students all reason as before, and tell him that his announcement is false and why. Everyone is surprised when he walks in on Tuesday with the test.

Did the teacher lie?

This version avoids some of the things people have argued about, such as the meaning of the word "surprise", or that the teacher chooses his day only after they pick one (if the logic was solid before he picks his day it is also solid after).

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Joe Pellino on Jan 13th, 2003, 2:51pm
I have thought of yet another brilliant conclusion.  Say that the test really cant be on friday.  Well then using that resoning, the test cant be on thursday either because since it cant be on friday, if it hasnt been given by wednessday, they will know it will be a surprise.  so with friday and thursday eliminated, that leaves 3 days.  Now, since it cant be on thursday or friday, it obviously cant be on wednessday.  So that leaves monday and tuesday.  Now this goes until monday right.  I aggree with the students on their thinking no test.  But because they told the teacher that there was no test, that must be what they really thought.  The teacher though knew they werent expecting a test so he let one day pass, to make them think they were right.  On tuesday SURPRISE he pulls a test out totally unexpected.  It is a surprise test.  The students didnt expect it at all.  It is basicly a trick question. ;D  There you have it. Case closed

Title: Re: anyone know the answer to the pop quiz riddle?
Post by redPEPPER on Jan 13th, 2003, 4:28pm
If the teacher can surprise the students with a test on tuesday, that means there's a mistake in their logic (and yours).

Title: Re: anyone know the answer to the pop quiz riddle?
Post by grrarrgh on Feb 4th, 2003, 9:34am
Well, having read through the majority of this thread, I feel as if  Nicodemus and Archon, among others, have already addressed the question sufficiently--at least the original, core issue, since there are additional interesting tidbits along the way that do not directly impact the question. It seems that most of the continuing discussion is about improving the method used to explain the riddle/the psychology of
decision-making/problem solving, or meaning, such as the
"surprise definition" debate, which is also interesting in its own right.

Here are my 2 incremental attempt at improvement and one bonus question, the first of which I believe covers little new logical ground, but might be more convincing for certain people; my third point might actually be mildly original:

Firstly, the (correct) claims are properly stated as "IF it is wednesday (or later), AND there is no test yet, THEN it must be later in the week;" "IF
it is already thursday (or later) AND the students are definitely untested, THEN there can be no test on friday." However, they do not actually allow the logic used by the
students to be valid.  I think problem-solvers tend to overlook the first part of each conditional's antecedent -- it has to be a given day (or later). In other words, the glossing over of logic by saying, as the students do, that "the quiz can't be on Thursday, because we know it won't be on Friday," forgets that the "base case" of sorts actually gives us: IF it is fridayAND there has been no test, THEN it cannot be given on fridaysuccessfully. This seems trivial, but to link it back to the problem, the logic of the studentsinitially is fine: "if the quiz doesn't happen by Thursday, it'll be obvious the quiz
is on Friday." So, it can't be Friday. BUT, in each ensuing step, the antecedent aspect of each claim is forgotten, and we just end up lumping the statements together.

Secondly, it is not the case that it cannot be on Friday; it would just mean that if it got to such a point, it would be cancellable by the students (to read the problem strictly).  In other words, by reserving the ability to "give" a predictable and hence, cancellable exam, the professor preserves the ability to keep the students in the dark.  Let me be clear -- I think problem-solvers confuse "cannot give" with "gives a cancellable exam," since the latter often seems equivalent from the perspective of a student   ;)

Thirdly (this is partially an expansion on the earlier
poor definition point made by others), what exactly do the students have to do? Say, before class every day, the students guess that today is the test -- they'll always be wrong (or get to cancel the test), but they'll never end
up having a test that counts.  Isn't this ability the equivalent of being able to apply the student's "logic," and if so, doesn't that make the problem sort of silly, rather than mind-boggling?  I'd try to explain in more detail, but I have to run to a Business Chinese class, and isn't part of the fun of discussing puzzles generating new ones?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by poseur on Feb 4th, 2003, 10:39am
The answer is simple. The teacher may say "The test is going to be a surprise," if and only if he knows he's smarter than the students. If he knows exactly where their logic will stop, then he can always take it one step farther and surprise them. Let's say his students were so stupid that Friday rolled around and they still couldn't figure out the test would be today. Then Friday would be an acceptable surprising day even though logic rules Friday out. If they're just a little smarter, they could rule out Friday but not look that far ahead on Thursday. Then he could surprise them on Thursday, even though logic rules Thursday out. And if they rule out every day of the week, then he may go one step farther than their logic and place the test on any day and surprise them even though logic would rule that day out.
Now if his students were even smarter, then they'd have figured this out and they would have ruled Friday out a second time because, even though Friday has already been ruled out, if Friday comes along and they haven't had a quiz yet, then they should say it's going to be today, not because they think it is but because it can't hurt to say it. And they could go on to rule out Thursday a second time if they know the teacher might think that way. But as long as the teacher is always one step ahead of their logic, he can always make the quiz a surprise.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by poseur on Feb 4th, 2003, 11:20am
And one other thing that makes it simple is, I would assume that they'll only be allowed to guess once. They can't come in Monday and say It's going to be today, and then come in Tuesday and say It's going to be today, and so on. Therefore, even though if Friday rolls around and they haven't had the quiz yet and they haven't guessed yet, then they can definitely cancel the quiz, the fact is that they don't dare wait that long before making their guess or the quiz will probably come while they're waiting. So Friday makes a very good day to put the quiz. It's true, when it comes it won't be a surprise, but by then they will have used up their one guess so it won't matter. Besides, he doesn't say exactly what the surprise will be. Maybe they won't be sitting there saying, "What? A quiz today?" But they will be surprised in the sense that they still can't believe he scheduled it for Friday.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by nietisne_ on Apr 21st, 2003, 5:52pm
what i say has been already said.i just think that people didn't payed the right attention to it.I think that the logic of the students disagrees with it self.if there was only one possible day that the quiz would be a surprise quiz(impossible) then this would be the day that we r looking for.this day appears for a moment to be monday(just the moment before we decide that it can't also be monday :)).But by the time we conclude the day is not monday we close a circle.Our conclusion now is that we should not expect the quiz on any day.in other words whenever he gives us the quiz the quiz will be unexpected.

and just a comment to a previous post :when u r setting the rules, then the only thing u have to do to be ahead of the logic of the players is to have no logic in what u do. :'(

Title: Re: anyone know the answer to the pop quiz riddle?
Post by jackper on Jun 4th, 2003, 12:53pm

I think the problem with the students argument is that it rests on two assumptions (at least), the truth of which can't be determined until after Friday, and therefore relate to the student's ability to 'know' the day of the test.

The assumptions: 1/ that the teacher isn't a liar, and 2/that the teacher isn't an idiot. (mighty risky assumption, judging from my experience).

Consider:
Thursday morning comes. The students say, 'well, he can't give the test tomorrow, so he has to have scheduled it for today.' They go to the teacher and say, 'Ok, you're giving the test today, because you can't give it tomorrow.' The teacher says, 'Ooops, I'm an idiot. Hadn't thought of that. I scheduled it for tomorrow.'

[So, if the students allow for the possibility the teacher is an idiot, then, on Thurday morning, they can't be sure the test will be Thursday, and so they can't eliminate Thursday]

You see, the students (on Thursday) can only know the test is on Thursday if they know the teacher's not an idiot (or a liar). How can they be sure he isn't an idiot??? They cant!...They have to wait till Friday to find out if he's an idiot.

The riddle involves the students 'knowing' something... and this means that the 'assumptions' they make are part of the essential riddle and cant just be assumed to be true by us.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jun 4th, 2003, 4:17pm
Okay - now how does that fit with the variant form I gave in my previous post? Would you say in the situation I described that the teacher was lying? Yet the students failed to predict when he would give his test, just as he said. The fundamental thing here is that the teacher is correct in his statement. How is he then a liar or an idiot?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by jackper on Jun 5th, 2003, 1:21pm

on 06/04/03 at 16:17:57, Icarus wrote:
 Would you say in the situation I described that the teacher was lying?

No, I would not say the teacher was lying.

on 06/04/03 at 16:17:57, Icarus wrote:
 The fundamental thing here is that the teacher is correct in his statement. How is he then a liar or an idiot?

No, the question isn't whether the teacher actually IS an idiot.
What's critical is that the students can't know whether or not he is an idiot. Even if we stipulate in the riddle that the teacher is not an idiot, the students still can't know this. The only way they can know if he is an idiot is to wait and see what day he picks for the test.

About the teacher being 'correct' in his statement, I'm not sure... suppose the teacher walked in on the last day of the week with the test, would you say his statement was correct then?

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jun 5th, 2003, 6:30pm
Yes, I would say that the teacher was correct - because the students did not predict he would give the test on friday.

When friday morning comes, the students are not expecting the test because they predicted there would not be a test at all.

(I suppose that at the start teacher should inform the students he will not tell them if they are right. They will only found out when test happens, or doesn't happen. And he should allow them only one chance - to avoid the "predict it every day" strategy.)

My point is that the flaw here is not that the students don't know if he is a liar or an idiot. Because he acts contrary to their prediction without being either. If the students had told him, "You are either lying, or mistaken, or you cannot give a test next week", and the teacher gives the test on any day, he will have shown their claim to be false. The teacher's statement was true. They failed to predict when he would give the test. So he wasn't lying, or mistaken.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by jackper on Jun 6th, 2003, 8:23am

on 06/05/03 at 18:30:34, Icarus wrote:
 When friday morning comes, the students are not expecting the test because they predicted there would not be a test at all.

Yes, I suppose the students can't know, on Friday, that there will be a test (or that the teacher will be trying to give them a test...they need to raise their logical argument before he pulls out the test) on Friday.... but, again, I say this is only because they cant know if the teacher is a liar, an idiot, or maybe he's just senile and forgot.

We both seem to be repeating ourselves without much convincing going on. Hopefully, someone else can step in here and help.   :P

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jun 6th, 2003, 5:38pm
Alright - try it this way: The teacher has a locked case with two keys. When he gives his proposal to the students, he tells them that he has written down when (if ever) he is giving the test on a sheet of paper. He then puts the paper in the case (a simple case - no magic tricks!) and locks it. He keeps one key, and gives one to the students. He tells them that when he gives the test, if they do not predict it, they will open the chest and see that this was his plan all along, or the test is canceled.

As before they predict that he cannot give the test. As before, on a certain day (it does not matter which) he walks in with it. They open the chest, and guess what? That day is written on the paper!
He did not forget (he could not forget, since it was written down beforehand). He was not mistaken, nor does he lie. Everything he said was true! They were unable to predict the day of the test.

If the students did not know if their professor was dependable, then the puzzle breaks down, because in that case, the student can't make their prediction. It is only because the students know that he IS dependable that they are able to reason that he cannot give it on any day. But once they have reasoned this, he can give the test whenever he wants completely in accordance with his words! The professor can be sharp as a tack and as honest as the day is long, and the students can know this in their bones, and to whatever other cliches are out there. Yet the puzzle STILL works!

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Leonid Broukhis on Jun 6th, 2003, 11:10pm
Let's rephrase the problem using a familiar item:

I throw a die. You try to guess the number thrown using the following scenario:
--> I let you make your guess or pass.
If you guess correctly (your guess does not necessarily have to be 1, it is equivalent to waiting until the corresponding round but speeds up the game), you win.
If you guess incorrectly, you lose.
If you pass, and the number thrown was 1, you lose, otherwise I say "It is not 1 ...", and the second round starts (replace "1" with "2" for the second round, go to --> ), and so on up to round 6, if you're patient enough and lucky enough, in which case you win by default.

What is your best strategy and the probability of winning?
No matter what is the answer, moving the problem into the probability domain lifts the paradox (die throws yield "surprizing" results by definition).

The usual train of thought, converted to the die domain, would go like this:
As there is a chance for you to win by default, to prevent that I'll have to use a loaded die that never goes 6 up, but here the similarity stops: you don't know whether I'm using a fair die or a loaded one, and you have to make an explicit guess of 5 -- winning by default does not move to the preceding rounds.

So, going back, the solution to the paradox is: the professor takes a risk. Simple as that.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by jackper on Jun 7th, 2003, 11:05am
Icarus,
Great! We are SO close to agreement.

In particular, in your last paragraph,
>>>>
If the students did not know if their professor was dependable, then the puzzle breaks down, because in that case, the student can't make their prediction. It is only because the students know that he IS dependable that they are able to reason that he cannot give it on any day. But once they have reasoned this, he can give the test whenever he wants completely in accordance with his words! The professor can be sharp as a tack and as honest as the day is long, and the students can know this in their bones, and to whatever other cliches are out there. Yet the puzzle STILL works!
>>>>
I would completely agree if you would agree to change the word 'know' to the word 'assume'.

Let me try once more to explain why I want this change:

First, the riddle is still a seeming paradox, the teacher can still give the test, unexpectedly on thursday, and the students logical argument is still fallacious if they simply assume the professor is 'dependable'.

Secondly, I still don't think the students can (theoretically as well as practically) 'know' the teacher is dependable. They can know the teacher HAS BEEN dependable in the past, but they cannot know he WILL BE dependable in the future. ('Future' being anything they do not yet have knowledge of, such as which day he picked.)

I suggest saying someone is 'dependable' is a bit like the old line about an 'irresistable' force meeting an 'immovable' object. If we instead ask 'what happens when an "up till now" irresistable force meets an "up to now" immovable object?', the mystery disappears.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jun 7th, 2003, 2:46pm
As the puzzle stands, the students can't "know" that professor is dependable. But you can adjust the situation again (such as with the lockbox) so that the students need not know, because the professor leaves no space for himself to be undependable. Leaving out the question of whether the students can know such a thing, even if they did know it, not just assume but actually KNOW, that the professor is dependable, they still get caught. Knowing the professor is dependable does not lead them away from their conclusion. And the professor can still flout their logic without being undependable in the slightest.

Leonid - I've scanned your post but will need to think on it a bit more before I respond. (I didn't want you to think that I was ignoring your post.)

Title: The definition of "know"
Post by Dean Foster on Jun 10th, 2003, 7:47am
I think it relies entirely on a good definition of "know."  This has bugged philosophers for years.  "Justified true belief" and all that.

But, if you take a statisitical/probabilistic approach, then knowledge is easy to define.  Can you profitable bet.  In this setting, there is a way of being sure that you will have shown a profit at the end of the week.  This doesn't require favorable odds (like 1/5), but instead can be done at any odds the professor wants to set.

So rephrase the problem as follows.  The professor says that she will accept bets at 1000 to 1 on whether the quiz will be held.  ONLY if the studen is very sure that he will win the bet should he place the bet.  Further the student has to bet that the quiz will happen.

Exercise: Show that the student can set it up so that either the student has made money at the end of the week, or the professor has lied.

See the following page for the solution:
http://gosset.wharton.upenn.edu/~foster/rants/a_surprise_quiz.html

later,

dean

Title: Re: anyone know the answer to the pop quiz riddle?
Post by towr on Jun 10th, 2003, 8:08am
there is no way to predict the unpredictable..

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jul 7th, 2003, 6:36pm
Try this scenario:

The professor walks in one day with five cards from an ordinary deck. The professor tells the class that he has chosen the cards purposefully. He then offers them the challenge: He will turn over the cards one at a time. Before he turns over the card, they will be given the chance to predict that the card is the Queen of Spades. If he turns over the Queen without them predicting it, or if they predict the Queen incorrectly, they will have a test next week. If they correctly predict the Queen, or make no prediction and the Queen is not one of the cards, then they will all be given automatic 100% for the test. He also tells them that he has chosen the cards in such a way that he is confident there will be a test.

This is equivalent to the situation in the cleaned-up version of the puzzle. The cards represent the 5 days, the day of the test is the Queen.

The students' reasoning is: Assume he has chosen the queen as one of the cards. If the first 4 cards go by without the queen, then it will be the last card and we can predict it. He knows this, so he won't put it in the last card. Since it can't be the last card, if 3 cards go by without a queen, we can predict that it is the 4th card. He knows this too, so it won't be the 4th card. Likewise, it can't be the 3rd card, the 2nd card, or the first card! Therefore he must not have chosen a queen at all.

The students therefore make no prediction, and are disappointed when the Queen shows up.

I think that this version makes the flaw in the students reasoning more obvious. It is not a matter of undependability in the professor, nor is it really a matter of the meaning of the word "know" (though the link Dean Foster provides is interesting and a valid point on its own - it is not the crux of the matter here) or of the word "surprise".

And while the professor obviously takes a chance (after all, they might have gotten it right if they simply took a  guess), he hedges his bet by encouraging them to attempt "predicting the unpredictable".

There are 2 fallacies in the students' reasoning. First is some circular logic: The students deduce that it will not be the last card by assuming it does not occur earlier in the deck. They then use this conclusion to show that it cannot occur earlier in the deck. Thus their conclusion that it cannot occur earlier in the deck is dependent on the assumption that it will not occur earlier in the deck.

This is the dope-slap they can give themselves if the queen turns up as one of the first 4 cards.

Now suppose they realize this after the initial session, but are unsure what to do about it. They sit tensely, not making a sound as the professor turns over the first 4 cards. RELIEF! None of the cards are queens! Their conclusion has played out, even though it was unjustified. They huddle for a final conference before the fifth card has turned over: "If there is a queen, then it must be this card. Since we can predict it, he won't have made it a queen either!"

The fallacy here is clear: Their conclusion that they can predict the queen's presence is based on the assumption that they know the queen is present! This same assumption is also in their original reasoning.

Both fallacies are also present in the original puzzle, but the second is well hidden. Since the professor announces there will be a test, it at first appears that this much they can assume. But note that their conclusion is that there is not a test - invalidating this assumption for their logic. If they had taken the professor at his word, they could indeed eliminate the possibility of a test on Friday by predicting it on that day. But by deciding that the professor was lying, they fool themselves into thinking that the prediction is not needed to avoid the test. Since they do not make it, they fail to eliminate Friday as a test day.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by towr on Jul 8th, 2003, 1:54am

What they do is say
"let's assume the queen is the last card,
if that is the case then the first 4 cards won't been the queen, and at the 4th card we'll be able to predict it is the 5th card.
Since we can't predict it (that's what the professor said) this can't be the case, thus our assumption that the fifth card is the queen can't be valid"

I don't know what it's called in english but here we call it "bewijs uit het ongerijmde", assume the things you want to disprove, and show that it leads to a false conclusion and thus said things cannot be true.

It's not circular, and effectively eliminates the fifth position. After that you can do the same for the other positions. It is imo in no way circular, but hinges on the truth of the professors claim they can't predict which card it is.

I think it's quite simple, if they consider it truthfull that they can't predict the queen, then they can't, because it is in direct contradiction.
Even if the professor would say, the fifth card is the queen, but you can't predict it. They would think, "hmm, it's the fifth card, but if we say it's the fifth card we would have succesfully predicted it, and the professor said we couldn't so it can't be the fifth card.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Leonid Broukhis on Jul 8th, 2003, 7:35am
The professor is truthful in saying that the students cannot predict it, but nothing prevents then from guessing. Nobody can predict the outcome of a lottery, but some lucky people do win.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by towr on Jul 8th, 2003, 8:36am
I don't see why one wouldn't be able to predict the lottery.. I predict I don't win. And I'm mostly right.
It's just a matter of applying probability theory, or instinct..

If the professor allways did the exact same thing he would be easy to predict.

I'd agree a prediction has to based on something other than wild guessing, but it needn't allways be right.
And I'd also agree that in this case the students can't predict with any accuracy over even chance. But unless they assume they might they won't even try and find out.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Icarus on Jul 8th, 2003, 5:21pm

on 07/08/03 at 01:54:14, towr wrote:
 I don't think I agree with your assesment about circular logic.What they do is say"let's assume the queen is the last card,if that is the case then the first 4 cards won't been the queen, and at the 4th card we'll be able to predict it is the 5th card.Since we can't predict it (that's what the professor said) this can't be the case, thus our assumption that the fifth card is the queen can't be valid"

You're right - put in this fashion it is not circular logic. It is incomplete logic, however. There are additional assumptions they have made, but failed to acknowledge. These are deadly in a "reducio ad absurdum" proof, since the contradiction only proves that at least one of the assumptions is false. If you have unrecognized assumptions, you will come to the wrong conclusion. In particular, the students assume that they will not predict an earlier card. That is, they assume not only that the queen is the 5th card, but also that they will guess - predict - know - whatever - it to be the fifth card. So the contradiction they find shows that either it is not the fifth card, or they will make a different prediction (or the professor will be wrong).

This now plays havoc with the next stage of their reasoning, because it requires them to predict the 4th card. Which means that they can no longer say that the fifth card is not the queen.

Quote:
 I don't know what it's called in english but here we call it "bewijs uit het ongerijmde", assume the things you want to disprove, and show that it leads to a false conclusion and thus said things cannot be true.

Some people may have special phrases, but I usually start off with something like "Suppose it's true...".

Quote:
 Even if the professor would say, the fifth card is the queen, but you can't predict it. They would think, "hmm, it's the fifth card, but if we say it's the fifth card we would have succesfully predicted it, and the professor said we couldn't so it can't be the fifth card.

...I've known some really stupid people, but I think most of them could figure out this professor is lying... ;)

Title: Re: anyone know the answer to the pop quiz riddle?
Post by towr on Jul 9th, 2003, 12:47am

on 07/08/03 at 17:21:11, Icarus wrote:
 In particular, the students assume that they will not predict an earlier card.

Actually they don't. If they have arrived at the 4th card (the 5th being then the only left) it follows from that they didn't predict any earlier cards.
Since they could then predict that the 5th card is it, and the professor said they couldn't they would conclude that they can't arrive at the 4th card without having made a prediction or having missed the queen.

I really think the problem lies in taking the professors statement as truth, rather than as something he beliefs which is not necessarily true. It might be true, but needn't be so they could succesfully predict the last card in this case.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by towr on Jul 9th, 2003, 1:42am
I've tried to formalize the riddle, and here's what I think it is.
http://www.ai.rug.nl/~towr/PHP/FORMULA/formula.php?md5=c138b8327dc1dafa1dd62b72c7649cd9

At each step Sn that the game isn't over (no predicted card and no queen) the predicted card is still to come and the queen is still to come.
And also the professor believes that the predicted card isn't the queen.

Depending on how you treat B you can get a contradiction when treating it like knowledge, or get nothing aside from that one of the cards is the queen and you'd be best of to just guess.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by mike1102 on Jul 9th, 2003, 7:35am
The reason Tuesday's test came as a total surprise is because the Professor had also given a test on Monday.

Title: Re: Pop Quiz Riddle
Post by Steve on Jan 5th, 2004, 12:54pm
What's wrong with the logic is that they think they know there is going to be no test next week, which would make a test on any day be a surprise.

Title: Re: Pop Quiz Riddle
Post by post@l on Jan 15th, 2004, 4:38am
This riddle reminded me a bit of Zeno's Tortoise paradox; The student's reasoning is logically correct, but in reality it just doesn't work that way...

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Jan 15th, 2004, 7:44am
Just considering the two-day version of this one, I came up with: If, after day 1, the test hasn't happened, then the pupils will be able to predict the test occuring on day 2, assuming they believe there will be a test at all (regardless of the test actually being scheduled). Having concluded that a scheduled test will not occur on day 2, on the ground that if it were, they would be able to predict it, the students then conclude that the test must be on day 1. But, realising that they believe the test must be on day 1, they conclude that the test cannot be on day 1 either and decide that they cannot believe there will be a test, as, were they to believe there will be one, then they can immediately predict it. The trouble is, while they have probability 1 of correctly predicting a test on day 2 if there is a test and they haven't previously predicted a test on day 1, in their argument that there can't be a test on day 1, they assume that there is 0 probability of there actually being a test on day 2, so that they will never find themselves in the position of predicting a test on day 2. Having made it an assumption that there will be no test scheduled for day 2, the students cannot be said to know that there will be a test on the second day, even if there isn't one on the first. They may well still claim to do so (since there's no penalty for error at that stage) but they would still be surprised to find one scheduled then.

Title: Re: Pop Quiz Riddle
Post by NO ONe on Jan 15th, 2004, 6:05pm

What is the flaw in the students thinkiing or if you want "logic"?

The flaw it seems is that they assess days closer to their current date in the same way they asses days that are further away.

In other words, Friday (which is further away) is cancelled before Monday which is closer and full of uncertainty.

The students falsely believe that Monday and Tuesday are as certain as friday and thursday.

The riddle also gives us the answer or what happens.

He gives the surprise quiz on Tuesday.

This part is harder to explain.

Title: Re: Pop Quiz Riddle
Post by ok on Jan 15th, 2004, 6:36pm
OK

I think I have the solution or reasoning behind this.

The problem with the students thinking is they destroy what they were given.

The Fact is that there will be a quiz.  Thats a fact!  You cannot change it!

The Students cannont guess what day it will be.  That part is impossible!  Yes impossible.  One student may be could guess but not all.

But this Riddle is more of a teaching tool.

For example If my students study for a quiz and predict that it will be in the first half of the week,  they will be prepared for it and I will cancel the quiz.  Just like the prof. in the riddle promised.

The students do not have to guess.  Knowing is easier than guessing!

In this riddle they Opened their mouths and guessed that that there was no quiz.  Once this was done, the students wasted their opportunity and a quiz was given.

If they kept their mouths shut, the professor would not know what the students would guess and he would have a harder time deciding when to give the quiz or he would not give it at all given the fact that his students were prepared for any quiz on any day at any time.

END OF STORY

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Jan 22nd, 2004, 7:57am
How about: "There will be a test tomorrow but you will believe there will not be a test tomorrow"

If you believe the statement, then you believe there will be a test, but believe that you don't believe it. If you are always aware of your own beliefs, then you will also believe that you do believe there will be a test tomorrow, but only if you believe that your beliefs are always correct will you believe that there will not be a test, and thus believe a contradiction. A more humble logician will simply have a (possibly) false belief that he believes that there will not be a test, and may or may not actually believe a contradiction. Of course, if you choose to beleive the statement false, then you believe "there will not be a test tomorrow or you will not believe that there will not be a test tomorrow" which can be verified even if there is a test tomorrow, provided you do not believe that there won't be one (which isn't necessarily the same as believing that there will be one).

The flaw in the Student's logic is (as stated previously by many people) that they derive a contradiction, then assume that the derived proposition is true while the premise that directly contradicts it, but which was used in the proof, is false, ie: they assume ((x[wedge]y)[bigto][lnot]x)[bigto][lnot]x where ((x[wedge]y)[bigto][lnot]x)[bigto]([lnot]x[vee][lnot]y) is correct

Title: Re: Pop Quiz Riddle
Post by jgray on Apr 5th, 2004, 2:57am
i believe that the answer lies in the wording of the riddle

"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."

He never says that you (the students) are going to have a suprise quiz. he says that we're going to have a suprise quiz.

leading me to believe that he has no idea when the quiz is.
He may have the power to cancel the quiz. but he may not be in control of when the quiz is.

Thoughts???

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Apr 5th, 2004, 7:25am
Yes, that does make a difference - they can no longer assume that the day was chosen to avoid there being a cancellation, so can't eliminate Friday... On the other hand, the fact the teacher states that there will be a test is still irreconcilable with his being truthful and the test being cancelled - if he doesn't know that the test isn't set on Friday...

Title: Re: Pop Quiz Riddle
Post by jgray- on Apr 6th, 2004, 2:34am
Good point.

He would have to know what day it is.

Still it is possible for it to be cancelled at the last minute still leaving the possibility that he doesn't know. Although it's highly unlikly

Ok consider this, the riddle ask "What's the flaw in the students' thinking?".
i think that one of the main flaws is that they are told that the test will be next week. therefore they have to choose a day of that week otherwise the professor is lying and if he is untrustworthy then we might aswell through logic out the door.

So maybe what they should of done is choose the most probable day for the test on.

So why is teusday the most probable day of the week for the test?.

This is also assuming that the professor went out of his way to make sure the students couldn't figure it out

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Apr 6th, 2004, 11:11am
I don't think "Tuesday" is significant - the professor could equally well have decided to hold the test on Friday, and the students would, having decided that he couldn't hold the test at all, be most surprised!

The point is that, despite the students concluding that there can't be a test and the professor is therefore lying about there being one, there is a test, and the professor's statement is completely true. Therefore, the students must have made a mistake since the facts flatly contradict their conclusion...

Title: Re: Pop Quiz Riddle
Post by jgray- on Apr 7th, 2004, 4:33am
yes i think thats all true

the students have applied logic to the problem but have still come unstuck

perhaps the students can't apply logic to this problem.
The reason being the dictionary definition of a surprise is:
To encounter suddenly or unexpectedly; take or catch unawares.

im not sure how to frase this but isn't there unexpected variables that come in to things all the time. Like a test flight of some plane and a bird flies in to one of the engines. (the pilot can turn round and say then "that was a surprise wasn't it" HEHE, sorry bad joke.) they can logically hypothisis about how the whole flight is going to go from take off to landing even take an educated guess as to the results but the probably didn't take in to account some dumb bird looking for a bite to eat.

anyway what im trying to get at is that the professor did say that it would be a surprise quiz and if you take the dictionary definition of surprise then no logic can be applied because its encounter suddenly or unexpectedly the professors trying to catch them out.

Also this leaves open the notion that the teacher still hasn't planned what day to have the test. (picking the most surprising moment)

oh one more thing it does say near the end of the riddle that the professor gives the quiz, totally unexpected!

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Apr 7th, 2004, 5:51am
You can rephrase the riddle in a way that removes all mention of surprise by the professor, and still have the students surprised when the test comes around. If you represent the test being on a given day as T{i} and the students predicting the test on a given day as P{i} where i ranges from 1 to 5, then the professor's statement is equivalent to:
[exists]!i[epsilon]{1,2,3,4,5}((T{i}[wedge][lnot]P{i})[wedge]([forall]j[ne]i([lnot]T{j})))

ie: there exists a unique i in the set 1 to 5 such that there will be a test on day i and the students will not predict that the test will be on day i and, furthermore, for any other day, j, there is not going to be any test on that day.

The essence of the problem still works if you restrict it to a single day - if the professor says "there will be a surprise test tomorrow" then that's equivalent to, letting B(X) represent belief in a proposition, X, "T[wedge][lnot]B(T)" or "test and not belief in test."

Applying the students' logic: assume the professor is telling the truth, then, tentatively B(T[wedge][lnot]B(T)) holds, which implies (B(T))[wedge](B([lnot]B(T))), giving (assuming awareness of belief) B(B(T)) and B([lnot]B(T)) - that is the students believe both that they believe there will be a test, and that they don't believe there will be a test. That is, believing the professor leads to believing a contradiction. Therefore, in order to avoid believing a contradiction, and thus believing themselves inconsistent, the students must disbelieve the professor. Where the wheels fall off their argument is that rather than believing [lnot]T[vee]B(T) - there won't be a test, or they will believe there will be a test (or possibly both) they ignore the possibility that the professor was lying about their beliefs and assume he was lying about the test and believe [lnot]T on its own (meaning that logically, they should come to believe the consequence, [lnot]T[vee]B(T), but that's beside the point) - at which point B([lnot]T) is true, and to avoid believing a contradiction, [lnot]B(T) must also be true - meaning the students reach the conclusion that there will be no test. If there is then a test, the professor was entirely correct. If the students instead believe T, then B(T) follows, and the professor's statement is false - the students believe there will be a test, so if there is a test, it won't be a surprise.

[e]formatting[/e]

Title: Re: Pop Quiz Riddle
Post by jgray- on Apr 8th, 2004, 1:23am
ok so u can change the riddle to make it so there is no surprise.

but i think tht the key word is Surprise.

i won't pretent that i understood the math that u present, far from it.

But u cant take human nature in to account,
it's a surprise quiz. like a surprise birthday sprung by your husband/wife. It's UNEXPECTED and u probably won't see it coming.

if the professor took all the math in to account. and assuming the students were smart enough to also do the math then that rules out all chance of surprise.

Also if there is an equation to how to figure out there will be a test next week then there also must be a formula to work out why he choose teusday.

please excuse me if i miss understand what u are trying to say.

But the dictionary meaning of surprise i believe still stands.
the professor said it would be a surprise i don't think it matters if the students get a surprise if the professor doesn't mention it

Title: Re: Pop Quiz Riddle
Post by Nigel_Parsons on Apr 8th, 2004, 4:00am
With mentions of "Jack of Spades" & "Queen of Spades" in several posts an answer becomes clear.

On the day the Prof announces the test he places (face down) in his desk the test papers and the four aces from a pack of cards (without mentioning this to the students!) He shuffles the four cards, and at the beginning of each day the following week, turns over one card. If it is the Ace of Spades he hands out the test. If not he discards that card.

In this manner the day of the test is a surprise (even to the Prof) thus cannot be accurately predicted by the students who have even less info to go on than the Prof, who will only know for certain once he turns the Ace of Spades, or once he has turned Wednesday's card and still not produced the Ace of Spades

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Apr 9th, 2004, 5:47am
But, since the students can make their prediction at any point, using a random system to pick the day of the test means that the professor can no longer guarantee it will be unexpected.

A rewording that removes mention of surprise: "I will give you a test next week. You have one chance to avoid the test - by telling me by the time classes start on the day of the test which day the test is."

[e]And, looking at the question of Tuesday: if you edit the original problem statement and replace "Tuesday" with any of "Monday", "Wednesday", "Thursday" or even "Friday", then does it make any less sense that the students, who concluded that there could be no test, are surprised when the test happens on whichever day?[/e]

Title: Re: Pop Quiz Riddle
Post by grimbal on Apr 27th, 2004, 1:53pm
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.

Title: Re: Pop Quiz Riddle
Post by wat_is_dis on May 19th, 2004, 3:57pm
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?

Title: Re: Pop Quiz Riddle
Post by Three Hands on May 19th, 2004, 4:14pm
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...

Title: Re: Pop Quiz Riddle
Post by rmsgrey on May 20th, 2004, 3:39am
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.

Title: Re: Pop Quiz Riddle
Post by Heya Gosper on May 20th, 2004, 5:50am
Urgh!

If you are so effing sure that it can't be on Friday then I won't cancel the test.

(Its on Friday).

Title: Re: Pop Quiz Riddle
Post by Hazy H on May 20th, 2004, 5:58am
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

Title: Re: Pop Quiz Riddle
Post by Three Hands on May 20th, 2004, 10:31am
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...

Title: Re: Pop Quiz Riddle
Post by GUEST on Jul 4th, 2004, 3:13pm
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)

Title: Re: Pop Quiz Riddle
Post by Kedirech on Jul 5th, 2004, 9:47pm
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  ::)
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  :P

Title: Re: Pop Quiz Riddle
Post by Three Hands on Jul 6th, 2004, 6:52am
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.

Title: Re: Pop Quiz Riddle
Post by mattian on Jul 6th, 2004, 5:42pm
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.

Title: Re: Pop Quiz Riddle
Post by mattian on Jul 6th, 2004, 5:50pm
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.

Title: Re: Pop Quiz Riddle
Post by elheber on Jul 30th, 2004, 8:58pm
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.

Title: Re: Pop Quiz Riddle
Post by rynoman190 on Aug 5th, 2004, 12:31am
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.

Title: Re: Pop Quiz Riddle
Post by guest on Aug 5th, 2004, 2:37pm
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."

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Aug 6th, 2004, 5:34am

on 08/05/04 at 14:37:53, 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...

Title: Re: Pop Quiz Riddle
Post by mattian on Aug 6th, 2004, 7:24am
Sheesh!

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Aug 7th, 2004, 12:33pm
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...)

Title: Re: Pop Quiz Riddle
Post by mattian on Aug 7th, 2004, 12:35pm
It wasn't the mathematics that baffled me - it was the English!

Title: Re: Pop Quiz Riddle
Post by jonderry on Aug 20th, 2004, 11:30am
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.

Title: Re: Pop Quiz Riddle
Post by daniducci on Aug 26th, 2004, 9:35am
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.

Title: Re: Pop Quiz Riddle
Post by towr on Aug 27th, 2004, 12:59am

on 08/26/04 at 09:35:43, 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.

Title: Re: Pop Quiz Riddle
Post by honkyboy on Sep 1st, 2004, 12:50pm
*(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.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Sep 1st, 2004, 2:56pm
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.

Title: Re: Pop Quiz Riddle
Post by honkyboy on Sep 1st, 2004, 10:52pm
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.

Title: Re: Pop Quiz Riddle
Post by Speaker on Sep 2nd, 2004, 12:28am
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.

Title: Re: anyone know the answer to the pop quiz riddle?
Post by Speaker on Sep 2nd, 2004, 12:48am
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 08/10/02 at 01:14:42, 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 testRyan 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.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Sep 3rd, 2004, 3:09am

on 09/01/04 at 22:52:49, honkyboy wrote:
 I never said that the students were convinced that there was not going to be a test on Friday.

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...

Title: Re: Pop Quiz Riddle
Post by Patashu on Sep 3rd, 2004, 4:12am
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.

Title: Re: Pop Quiz Riddle
Post by towr on Sep 3rd, 2004, 6:18am
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.

Title: Re: Pop Quiz Riddle
Post by Three Hands on Sep 3rd, 2004, 7:35am
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...

Title: Re: Pop Quiz Riddle
Post by Hobbes199 on Sep 14th, 2004, 11:19am
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.

Title: Re: Pop Quiz Riddle
Post by Kent on Feb 16th, 2005, 3:21pm
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!

Title: Re: Pop Quiz Riddle
Post by Tarzul on Mar 16th, 2005, 4:44pm
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.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Mar 17th, 2005, 7:32am
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.

Title: Re: Pop Quiz Riddle
Post by colin a on Mar 22nd, 2005, 4:13pm
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.

Title: Re: Pop Quiz Riddle
Post by Djinon on Apr 19th, 2005, 11:14am
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.

Title: Re: Pop Quiz Riddle
Post by Djinon on May 4th, 2005, 10:21pm
I posted the above around 100 thread views ago. Am I right or something? Need to have some feedback...

Title: Re: Pop Quiz Riddle
Post by towr on May 5th, 2005, 2:22am
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.

Title: Re: Pop Quiz Riddle
Post by Jonathan La Fratta on Jun 12th, 2005, 9:27pm
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

Title: Re: Pop Quiz Riddle
Post by paul schmitz on Jun 30th, 2005, 11:47am
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.

Title: Re: Pop Quiz Riddle
Post by Jujubeeks on Jul 6th, 2005, 2:39am
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...

Title: Re: Pop Quiz Riddle
Post by Brad Bida on Aug 27th, 2005, 12:08am
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!

Title: Re: Pop Quiz Riddle
Post by Nasta on Sep 3rd, 2005, 7:24pm
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 ... ??? ??? ???

Title: Re: Pop Quiz Riddle
Post by dhasenan on Oct 29th, 2005, 7:36pm
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.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Oct 30th, 2005, 5:11am

on 10/29/05 at 19:36:45, 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.

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.

Title: Re: Pop Quiz Riddle
Post by nnahas on Nov 18th, 2005, 1:03pm
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?

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 18th, 2005, 7:42pm
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. :o

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.

Title: Re: Pop Quiz Riddle
Post by towr on Nov 19th, 2005, 3:22pm

on 11/18/05 at 19:42:31, 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.
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)

Title: Re: Pop Quiz Riddle
Post by Eigenray on Nov 19th, 2005, 4:41pm
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).

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 20th, 2005, 7:11pm

on 11/19/05 at 15:22:02, towr wrote:
 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)

No - if a counter-example exists, the statement is false, whether or not anyone knows about it. If you just don't know about it, then the statement is merely undecided, not undecidable.

Undecidable statements are statements that can be neither proven nor disproven using the axioms and logic of the mathematical theory under consideration. As such, they cannot have counter-examples within that theory. An undecidable statement can be added to the axioms of the theory to create a new daughter theory in which the statement is true. And the negation of the statement can also be added to the axioms of the original theory to create a new daughter theory in which the statement is false. There are no counter-examples to either one.

on 11/19/05 at 16:41:55, Eigenray wrote:
 (although even if a universal statement has a counter-example, that doesn't mean it's provably false).

??? What definition of "universal" are you using that allows counter-examples?
Demonstrating the existance of counter-examples is exactly how you prove a universal statement is false!

Title: Re: Pop Quiz Riddle
Post by Eigenray on Nov 20th, 2005, 8:58pm

on 11/20/05 at 19:11:02, Icarus wrote:
 Demonstrating the existance of counter-examples is exactly how you prove a universal statement is false!

Well, proving the existence of counter-examples, sure.  But maybe x=17 is a counter-example to the (Pi4?) formula
For all x, Goldbach's conjecture is false.

Title: Re: Pop Quiz Riddle
Post by towr on Nov 21st, 2005, 5:10am

on 11/20/05 at 19:11:02, Icarus wrote:
 No - if a counter-example exists, the statement is false, whether or not anyone knows about it. If you just don't know about it, then the statement is merely undecided, not undecidable.
I'm pretty sure that's what I said..

Quote:
 Undecidable statements are statements that can be neither proven nor disproven using the axioms and logic of the mathematical theory under consideration.
I was under the impression that undecidability meant that there were true statements that could not be proven, and false statement that could not be disproven (by using axioms and applying rules).
Certainly there are numerous statements that trivially can't be proven, since they're contingent rather than tautological or contradictory. I wouldn't call those undecidable though.

Quote:
 As such, they cannot have counter-examples within that theory.
But then you're just looking at the syntactical side, not the semantical side. Undecidability only really means something when you look at both the model and theory; whether the theory is sound and complete with respect to the model.
Counter-examples are typically something found in a model (or at the very least corresponding to some entity in the model).

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 21st, 2005, 4:48pm

on 11/20/05 at 20:58:41, Eigenray wrote:
 Well, proving the existence of counter-examples, sure.  But maybe x=17 is a counter-example to the (Pi4?) formulaFor all x, Goldbach's conjecture is false.

If you can't prove it, how is it a counter-example? Possibly the (Pi4?) formula is false for x=17, but for the formula to be false, you have to be able to show it.  Otherwise the formula is undecidable or true, not false.

on 11/21/05 at 05:10:07, towr wrote:
 I'm pretty sure that's what I said..

My point was in answer to this statement:

on 11/19/05 at 15:22:02, towr wrote:
 However, if the statement is undecidable, you can't know if the statement does or doesn't have a counter example.

For a statement to be undecidable, it cannot have a counter-example. So if you know the statement is undecidable, you do know that is does not have a counter-example. If it has a counter-example, it is false, whether or not you know of the example.

on 11/21/05 at 05:10:07, towr wrote:
 I was under the impression that undecidability meant that there were true statements that could not be proven, and false statement that could not be disproven (by using axioms and applying rules).

If they cannot be proven, in what sense are they "true"? If a statement cannot be proven, then there exist completely consistent sub-theories in which the statement is true, and completely consistent sub-theories in which the statement is false. (I'm sure you understand what I mean, but for clarity's sake: a "sub-theory" of a mathematical theory is another theory containing all the axioms and primatives of the original, but with other axioms and primatives added.)

If the statement is somehow "true", how can I have a nice consistent sub-theory in which it is provably false?

Quote:
 Certainly there are numerous statements that trivially can't be proven, since they're contingent rather than tautological or contradictory. I wouldn't call those undecidable though.

I have no idea what you mean by "contingent".

Quote:
 But then you're just looking at the syntactical side, not the semantical side. Undecidability only really means something when you look at both the model and theory; whether the theory is sound and complete with respect to the model.Counter-examples are typically something found in a model (or at the very least corresponding to some entity in the model).

No - undecidability is often demonstrated by use of models, but it is only meaningful for the theory alone. It means exactly that neither the statement nor its negation can be proved within the theory. This is usually demonstrated by showing the existance of a model of the theory in which the statement is true, and of another model in which it is false.

Such statements are true or false in the model, but they are NOT true or false within the theory itself. There, "undecidable" is the most you can say. Models (there are many models of a theory) represent subtheories, with axioms beyond those of the theory modeled. So truth or falsity of a statement in the model does not imply the same for the theory.

A counter-example within some model is not a counter-example within the theory. Only when the counter-example exists within all models does it count as a counter-example within the theory itself.
________________________________________

To the particular point of nnahas' conundrum. His logic reduces to this:

(1) Conjecture is undecidable --> no counter-examples exist.
(2) No counter-examples exist --> Conjecture is provable.
(3) Conjecture is provable --> Conjecture is not undecidable, contradicting the assumption.
(4) Therefore, the conjecture must be decidable.

The downfall is step (2), which only holds for a different meaning of the terms than used in (1). To be more specific:
(1) Conjecture is undecidable --> no counter-examples within the theory exist.
For the conjecture to be truly undecidable, there must be models in which it is false (in particular, the theory with the negation of the conjecture added as an axiom is a model in which it is false). These models may have counter-examples within those models, but such counter-examples do not hold without the added axioms of the model.
(2) No counter-examples within any models--> conjecture is provable.
Stated as such, this is also true. If no model exists in which the statement is false, then in particular, the theory consisting of adding the negation of the conjecture as an axiom must not be a valid model. The only way it can not be a valid model is if it is contradictory. But if it is contradictory, then the conjecture must be provable in the original theory.

However, the weakness in the argument should be clear. The conclusion of (1) is not the same as the hypothesis of (2), and therefore the logic chain is broken.

Title: Re: Pop Quiz Riddle
Post by Three Hands on Nov 21st, 2005, 5:32pm
OK, I'm not entirely sure about the mathematical expressions used, so this may have been said already in ways I didn't quite understand, but the way in which Kant expresses the nature of moral imperatives (following a logical structure) seems to be a good way of showing contingency:

An imperative should be considered contingent if you could will that it apply for everyone, and could also will the negation of the imperative to apply to everyone. (e.g. "The lucky number people should have is 7" would be contingent - unless you are somewhat zealous about your belief in lucky numbers, but then you're just strange :P)

Contingency is also popular in possible world theories, where differences between worlds show what is contingent in a given world, while necessary things will remain constant in all possible worlds.

It seems that undecidability falls under the same kind of idea as contingency, in that just lacking a counter-example is not enough to prove a conjecture is true, as you also need to show that the negation of the conjecture has a counter-example. Otherwise, the conjecture is undecidable.

At least, that's how I understand contingency, and undecidability on the basis of this thread. Like I say, I could be wrong.

Title: Re: Pop Quiz Riddle
Post by Eigenray on Nov 21st, 2005, 8:20pm

on 11/21/05 at 16:48:06, Icarus wrote:
 If you can't prove it, how is it a counter-example? Possibly the (Pi4?) formula is false for x=17, but for the formula to be false, you have to be able to show it.  Otherwise the formula is undecidable or true, not false.

Oh.  I thought we were talking about counter-examples in the standard model, i.e., if I say "for all x, phi(x)", and there's some x0 such that phi(x0) is false, then that would be a counter-example, even though I might not be able to prove phi(x0) is false.

For example, suppose for the sake of argument that Goldbach's conjecture is undecidable.  In particular, it is not provably false, so there are no counter-examples (in the standard model) since in this case, any counter-example could easily be proved to be such.  That means it's true.  Now if I say "for all x, Goldbach's conjecture is false," then wouldn't x=17 be a counter-example to the latter statement?

On the other hand, for something like the continuum hypothesis, what exactly would a counter-example be?  A set x, in some model of ZFC, with |N| < |x| < |R|?  Or a proof in ZFC that there exists such an x?  The former exists, assuming Con(ZF).  The latter doesn't.

There's a standard model of arithmetic.  Any statement in number theory is either true or false in this model, and that's what people usually mean.  Especially if they distinguish between "true" and "provable".  For example, Goodstein's Theorem is true, even though it's not provable in PA.

But there isn't even a standard set of axioms for set theory, let alone a standard model.  So if someone asks "is foo true," you likely have to ask them what they mean by "true."

Does every field have an algebraic closure?  When asked outside the context of logic, the answer is yes.  Not because we write down a formal derivation from our favorite set of axioms, but because we just "prove" it, like any other theorem: it follows from other things that we know are "true".  But of course, this discussion is specifically in the context of logic.

Is the axiom of choice true?  Well, how did all the other axioms get in ZF?  Because they're necessary for doing mathematics.  A lot of mathematicians use AC (algebraic closures, maximal ideals, bases, ...), so they work in ZFC, where AC is true.  It's so widely accepted because (0) Con(ZF) -> Con(ZFC); (1) it's, well, "true": if you take a bunch of nonempty sets, their product is non-empty; and (2) It's so damn useful.  So yes, AC is true (in ZFC, which is usually assumed if no context is given).  (A similar argument could be made for Foundation: it's consistent with ZF \ Foundation, and it should be true, but it's not really necessary, since most math is done in WF anyway.)

On the other hand, is there an uncountable set smaller than R?  It's consistent either way, but (1) there's no intuitive answer, and (2) it doesn't really matter to most people, other than logicians.  In this case, there's no answer to the question "is CH true," because when asked with no context, any answer would stay within ZFC, where it's undecidable.  But if you were to ask, say, "is CH true in ZF+(V=L)", then the answer is yes.

Finally, it should be noted that I don't really know what I'm talking about, and I'm pretty much just rambling here.

Title: Re: Pop Quiz Riddle
Post by towr on Nov 22nd, 2005, 3:17am

on 11/21/05 at 16:48:06, Icarus wrote:
 For a statement to be undecidable, it cannot have a counter-example.
Not in the theory, no, but in a model yes. As you say yourself sometime later.
I think you're not understanding entirely where I'm coming from. I'm working a lot with modal logic at the moment, and the theories there are formed after the model, because meaning is important. And without a model you have no meaning; it'd just be symbol manipulation.
The theories there are proof systems to reason about the model. The same goes for most logics afaik.

Quote:
 If they cannot be proven, in what sense are they "true"?
They are true in they're meaning. If our 'world' (universe) contains God, then  he's true, even though we can't proof him with all the laws and rules we created to reason about the world.

Quote:
 I have no idea what you mean by "contingent".

tautology: A | ~A
contingency: A
Whether it's true or not depends on what value A gets.
Unlike the other two examples. It doesn't make sense to use the first two as axioms, because adding a tautology adds nothing, and adding a contradiction makes everything follow from the theory.
Adding a contigency as axiom can create a 'meaningfull' subtheory. (Although not really in this case)

Title: Re: Pop Quiz Riddle
Post by nnahas on Nov 22nd, 2005, 12:21pm

on 11/21/05 at 16:48:06, Icarus wrote:
 The downfall is step (2), which only holds for a different meaning of the terms than used in (1). .

I fully agree with your point of view. Do you think some analogy can be drawn to different meanings of the word "Predict" in the pop quiz riddle ? I have a  feeling that the meaning of the word Predict is changing as the pop quiz story develops, although I still can't pinpoint exactly what are the different meanings

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 22nd, 2005, 5:01pm
So many controversies, so little time... ::)

Starting in order:
Three Hands, thank you for your comments. I don't want to appear to ignore them, but just as you have trouble understanding the deeper mathematics, I don't really follow your deeper philosophical statements, even if both are drawing on the common well of Aristotlean logic. The terminology is not completely co-incident.
_________________________________________

Eigenray - my comments to you have nothing to do with any model. I was speaking of matters strictly within the theory itself (I only included the model stuff as I came to realize what towr was refering to).

If it cannot be proven that 17 is a counter-example, then 17 is not a counter-example (this is of course a little more general than saying that you or I cannot prove it - I mean that a proof does not exist). But even if there are no counter-examples - or more generally, that a conjecture cannot be shown false, this does not mean the conjecture is true. To be true (within the theory), it must be provable. So a lack of counter-examples or counter-proofs just means the conjecture is either undecidable or true.

on 11/21/05 at 20:20:36, Eigenray wrote:
 There's a standard model of arithmetic.  Any statement in number theory is either true or false in this model, and that's what people usually mean.  Especially if they distinguish between "true" and "provable".  For example, Goodstein's Theorem is true, even though it's not provable in PA.

Really? Interesting that I have never heard of this model before. Obviously Godel never heard of it, either. Which is really too bad, since it would have stopped him from making a fool of himself by very famously proving that such a model is impossible. What is this model? How is it defined? How does it avoid the logic of Godel's incompleteness proof?

Also, what is "PA"?
_________________________________

towr - I don't know about the modal logic you are dealing with, but you have it completely backwards for how mathematics works.

Perhaps some definitions would help here. You surely know by now how much I love defining things!:D The definitions below are a simplified version of the formalist view.

A Mathematical Theory consists of:
• A list of symbols, called the "Primatives".
• Rules for constructing "Statements" and "Objects" from the primatives.
• A finite collection of rules for specifying certain statements. The rules are generally called by different names depending on the particular theory and their nature. Bourbaki refered to his as "Axiomatic Schemas". The statements specified by the rules are called "Axioms". The reason for schemas rather than simply a list of axioms is because the axioms cannot generally have statements as values for their variables, without running into problems with self-reference and inducing contradictions. So such "axioms" as (A implies (A or B)), must instead be stated as schemas, with all possible replacements for A and B providing different axioms.

The theory is developed by
• Adding "Definitions": additional symbols and statements relating them to the primatives, which can be thought of as more axioms, and
• Creating "Proofs", ordered lists of statements in which each entry is either an axiom, or is a statement B preceded by two other statements of the form "A" and "A implies B". A "Theorem" is any statement that occurs in one of these proof.

Proofs as we ordinarily think of them are, to the formalist, merely a "meta-argument" (one outside the theory, about the theory) that a formal proof as defined above exists. Statements are "true" in a theory if they are theorems (i.e., they occur in some formal proof). Statements are "false" if their negation is a theorem. If neither a statement nor its negation are theorems, then the statement is undecidable.

If T and T' are theories such that every primative in T  corresponds to a primative, statement, or object in T', so that under this correspondence, every statement or object of T becomes a statement or object of T', and every axiom of T is a theorem of T' (note that all axioms are also theorems within their own theory), then T' is called a "subtheory" of T. If a statement is true in T, then its corresponding statement in T' is guaranteed to also be true. This is because the proof in T corresponds to a list of statements in T' that can be expanded to a full formal proof by preceding the list with proofs of all the theorems used in it without precedents.

T is called contradictory, or inconsistent, if there is a true (i.e., provable) statement within it whose negation is also true.

A "Model" of T is simply a subtheory of T. Usually the model is a construction within some other theory S, (of which it is necessarily also a subtheory). Models are most useful in demonstrating relative consistency, i.e. in showing that if S is consistent, then so is T.

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 22nd, 2005, 5:30pm
Continuing in a second post to sneak around the evil gremlin that limits vebosity...  >:(

Note that from a mathematical point of view, the theory is where the essential meaning of what we are talking about is found. For instance, group theory consists of set theory (your pick which one) along with the added primative G (the group), * (the operation), and e (the identity), and the axioms:

(1) for all x in G, e*x = x*e = x
(2) for all x, y, z in G, x*(y*z) = (x*y)*z
(3) for all x in G, there exists y in G such that x*y = y*x = e.

The meaning of groups, all that we need to discuss and develop the idea of group, is found in these. Anything beyond is not about groups, but about specific groups or collections of groups.

In particular, models such as the integers, do not give any meaning to groups in general that is not already found in those axioms. The only thing that the existance of integers as a model for groups tells us is that our group theory is at least as consistent as arithmetic.

Your concept seems to be that you have one underlying model with many theories built off it. "True" and "False" are defined in terms of this model, and not by the theory. But in mathematics, this is not how we think of it. In fact, theories have many, many distinct models (even complete theories do, though they do not differ in any way discussable within the theory itself). What is true within one model, if not provable in the theory, will be false within other models.

This is why I do not, and cannot, define "true" in terms of what is true in some underlying model. Which model do I choose? Why should this model be chosen over others where the same statement is false? If we define true by some underlying model, then we are studying the model, and not the theory. It is fine to study the model, but we should not be tempted to think that results there always apply to the theory.

____________________________________

nnahas, the problem is indeed deeply rooted in semantics. I believe it has more to do with false assumptions the students develop from assuming the words have one meaning while the professor is interpreting them differently. I have explored this and posted it before, and am right now too involved mentally in other arguments ( ::) ) to be motivated to delve back into it at this time. Alas, if you want my opinion, you will have to find it posted on previous pages.

Title: Re: Pop Quiz Riddle
Post by towr on Nov 23rd, 2005, 4:33am

on 11/22/05 at 17:30:40, Icarus wrote:
 Your concept seems to be that you have one underlying model with many theories built off it. "True" and "False" are defined in terms of this model, and not by the theory. But in mathematics, this is not how we think of it.
Then obviously we're simply talking about different things.
Apparantly it means something different in logic than it does in mathematics. Which is quite acceptable to me..

Title: Re: Pop Quiz Riddle
Post by Eigenray on Nov 23rd, 2005, 1:40pm

on 11/22/05 at 17:01:41, Icarus wrote:
 If it cannot be proven that 17 is a counter-example, then 17 is not a counter-example

I guess I just don't like that terminology.  A counterexample is an example, a specific element of some model which makes a given statement false in that model.  A "counterexample in the theory" would be better called a "disproof".

The only misunderstanding here is that nobody bothered to define "counterexample" the first few times it came up (though in my opinion, the first time nnahas used it, he was clearly referring to a counterexample in the standard model of arithmetic).

Quote:
 Also, what is "PA"?

That's the million dollar question.  There I meant Peano Arithmetic -- the first-order system of arithmetic where the axiom of induction is replaced by a schema: one axiom for each arithmetical formula.

On the other hand, the [link=http://en.wikipedia.org/wiki/Peano_axioms]Peano Axioms[/link] is a second-order system: induction is taken care of by quantifying over properties.

The key difference here is that while there are nonstandard models of Peano Arithmetic, there is only one model for the Peano Axioms: the natural numbers.  This is what I mean by the standard model.  Even if you're talking about first-order logic, the natural numbers still exists as a model; it is the standard model because furthermore it satisfies the second-order induction axiom.

Quote:
 A "Model" of T is simply a subtheory of T.

A model is not a theory.  A theory T is a set of sentences.  A model M is a set (universe) together with an interpretation of the language L.  Truth in M is defined recursively: k-ary predicate symbols of L are interpreted as subsets of Mk; function symbols are functions; logical connectives work as you'd expect; similarly for quantification: "for all x, phi(x)" is true in M iff for all s in M, the formula phi(x), with every (unbounded) x replaced by s, is true in M.  (Part of the definition is that every sentence is either true or false in M: the theory of any model is complete.)  M is a model for T iff every sentence of T is true in M.

For example, the theory of groups is not a group; its models are groups.  Thus (G,*) is a group iff the axioms of group theory are true in G.

Quote:
 For instance, group theory consists of set theory (your pick which one) along with the added primative G (the group), * (the operation), and e (the identity), and the axioms:

You certainly don't need any of the usual axioms of set theory, or even the [in] predicate.  It's just the first order language with a binary function symbol and 3 axioms (a constant symbol for the identity is not strictly necessary as it is definable).

Title: Re: Pop Quiz Riddle
Post by Icarus on Nov 23rd, 2005, 7:31pm
Perhaps the terminology has changed from what I learned. For me, "The Natural numbers" are the theory, and are defined by the Peano Axioms or some equivalent formulation (whether first-order, second-order, or twenty-third order...). Models for the natural numbers are such constructions as the set { {}, [], [}, []}, ... } (i.e., 0 = {}, 1={0}. 2={0,1}, etc), or certain equivalence classes of pairs of line segments (which is effectively how Euclid develops them). [Yes, I am including "0" in the natural numbers here. Call them the finite Cardinal numbers, or the Whole numbers, if you prefer.]

As a second example, "The smallest algebraicly and topologically complete field of characteristic 0" is the short description of the Theory of Complex Numbers. The most common Model of the complex numbers is R x R, along with the appropriate definitions of + and *. A second well-known model is the quotient R[x]/(x2+1).

Part of the problem may be that I am only interested in such things for their practical application to mathematics as a whole, and not for the sake of the thing itself (not that I disapprove of interest for the sake of it- I am a mathematician, after all - but I do not share it). This is why I take the theory of groups to be a subtheory of the theory of sets. Leaving set theory out makes Group theory tepid and difficult to talk about. Put set theory in, and suddenly Group theory is robust and full of examples.

As for counter-examples: For me, if it is not within the theory but only within some particular model, then it does not align with any concept of what is meant by "counter-example" that I normally encounter. For example (lets choose something a lot simpler here than obscure and difficult conjectures), a counter-example to the statement "For all x, x = 1" would be the value 2. A particular value well-defined in the theory (not just some model of the theory) which shows that the statement in question is false. On the other hand, the statement "For all natural numbers x, if x>0, then 0 c x." is true for the set-theoretic model of the natural numbers I gave above, but is false or even non-sensical for other models of the Natural numbers. In particular, the fact that it is not true for the equivalence classes used in the Geometry model does not count as a "counter-example" to me. (A bad example, I know, since the element operator c is not a part of of the theory of natural numbers, so the statement is not valid, but hopefully you follow my meaning.)

Title: Re: Pop Quiz Riddle
Post by Alky on Dec 6th, 2005, 11:41am
I can't say I've read through this whole thread, but I have the solution to the riddle.

Essentially what the students have is an inductive proof. Each case obviously implies the next case as long as the base case is true.

It seems like everyone is focusin on the induction step (not friday gives not thursday) but ignoring the obvious fact that if the test happened on tuesday, the students wouldn't be able to know the test is on friday when thursday night arrived (because is happened on tuesday).

In short, the base case isn't proven.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Dec 7th, 2005, 8:53am

on 12/06/05 at 11:41:35, Alky wrote:
 I can't say I've read through this whole thread, but I have the solution to the riddle.Essentially what the students have is an inductive proof. Each case obviously implies the next case as long as the base case is true.It seems like everyone is focusin on the induction step (not friday gives not thursday) but ignoring the obvious fact that if the test happened on tuesday, the students wouldn't be able to know the test is on friday when thursday night arrived (because is happened on tuesday).In short, the base case isn't proven.

What exactly do you think the base case is?

Not Friday is proved by conrtradiction - were it to be Friday, then the students would know on Thursday (because it wouldn't have been Tuesday or any other day)

Title: Re: Pop Quiz Riddle
Post by Alky on Dec 7th, 2005, 2:11pm

on 12/07/05 at 08:53:40, rmsgrey wrote:
 What exactly do you think the base case is?Not Friday is proved by conrtradiction - were it to be Friday, then the students would know on Thursday (because it wouldn't have been Tuesday or any other day)

The base case for the induction is "the test won't happen on friday"

To know that the test won't happen on friday, you need to know it hasn't happened on monday, tuesday, wednesday or thursday. (This is made abundantly clear by the fact that it's thursday night and they would expect the test to be tomorrow)

The problem is that the logic goes something like this:

IF the test doesn't happen before friday
THEN the test can't be a surprise on friday
THEREFORE it won't happen on friday (IF the test doesn't happen before friday)
THEREFORE it won't be surprise on thursday (IF the test doesn't happen before friday)
THEREFORE it won't happen on thursday (IF the test doesn't happen before friday)
etc.

That's it. We assumed what we proved. Circular reasoning.

Title: Re: Pop Quiz Riddle
Post by towr on Dec 7th, 2005, 2:34pm
No, the reasoning is

If the test doesn't happen before friday, it must be on friday
But if it is on friday, it won't have been a surprise ; which is a contradiction, the test is a surprise.
Therefore the test must happen before friday.

Title: Re: Pop Quiz Riddle
Post by Alky on Dec 7th, 2005, 6:34pm

on 12/07/05 at 14:34:13, towr wrote:
 No, the reasoning isIf the test doesn't happen before friday, it must be on fridayBut if it is on friday, it won't have been a surprise ; which is a contradiction, the test is a surprise.Therefore the test must happen before friday.

No, it wouldn't be a surpirse if the test was friday and the test already hadn't happened on monday, tuesday, wednesday and thursday. It would be a surprise UNTIL thursday night.

Title: Re: Pop Quiz Riddle
Post by towr on Dec 8th, 2005, 2:26am

on 12/07/05 at 18:34:19, Alky wrote:
 It would be a surprise UNTIL thursday night.
It would be a surprise until you know can't happen on thursday or before. I doubt they have classes thursday night.

If the test is on friday, they will know at the end of thursdays classes that it will be the next day, hence it's no surprise. Thus it can't be on friday.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Dec 8th, 2005, 12:22pm

on 12/07/05 at 14:11:54, Alky wrote:
 The problem is that the logic goes something like this:IF the test doesn't happen before fridayTHEN the test can't be a surprise on fridayTHEREFORE it won't happen on friday (IF the test doesn't happen before friday)

But if the test does happen before Friday, then it won't happen on Friday either.

Therefore the test won't happen on Friday (If the test doesn't happen before Friday or the test does happen before Friday)

So the only way the test can take place on Friday is if it neither takes place before Friday, nor fails to take place before Friday.

Title: Re: Pop Quiz Riddle
Post by Alky on Dec 8th, 2005, 2:07pm
Alright, I concede that point.

Just one thing to think about: We don't even need any of this "friday then thursday" stuff because we can just use the 'surprise' premise to disprove any day we choose.

Assume we figured out the test was on tuesday. But wait! The test is surprise so we can't figure out it will be on tuesday.

I think the problem comes because the day we expect the test to be is not equal to the day the test really is. Assume the test IS on friday. We go through the logic and expect it not to happen: surprise! it's on friday anyways.

Title: Re: Pop Quiz Riddle
Post by towr on Dec 8th, 2005, 2:50pm

on 12/08/05 at 14:07:10, Alky wrote:
 Just one thing to think about: We don't even need any of this "friday then thursday" stuff because we can just use the 'surprise' premise to disprove any day we choose.
Yes, that's the main problem. If you believe it will be a surprise, you cannot believe you know when it is.

If the Prof says: "There is a surprise test friday".
You can't believe that this is true, even if it is.

Title: Re: Pop Quiz Riddle
Post by River Phoenix on Feb 21st, 2006, 2:06am
This statement of the problem is a little strange because you can take a guess and the professor won't hold the quiz if you're right. Fact is the professor took a risk: suppose the students play the strategy that they never guess until Thursday night rolls around, then if the quiz hasn't been given yet they guess that it is on Friday. In either case - if there is no quiz at all, or if it was planned for Friday, then the students know that there will be no quiz on Friday. So the professor should never put the quiz on Friday because the students might play this strategy. So in this version, the flaw involving not knowing that there will be a quiz at all, is not actually important; it seems to me that it all just boils down to the problem being a logical fallacy (If we can prove X, then not X).

Title: Re: Pop Quiz Riddle
Post by River Phoenix on Feb 21st, 2006, 2:17am
Basically, there are number of assumptions that are taken as axioms. Together, they lead to a contradiction. If we continue to take all of what is in the problem as axioms, then we conclude that the problem is simply nonsense. Otherwise, we have to remove one of the axioms. There is no particular one of the axioms that is wrong, I just find it easiest to say that in fact it was the professors statements that were actually just assumptions.

In many statings of the paradox (in this one too if you disregard my previous post, which I'm not too sure on actually), you could simply say that the assumption that the test will be given in the week, is false. It's negation is that the test might not be given in the week. This suffices, since the test will always be a surprise, after all it might not be given at all. We simply can't assume that the test will both definitely be given and will definitely be a surprise, since this is obviously not true once Friday morning rolls around. So either the test won't necessarily be given, or the teacher was wrong and took a risk - meaning the test won't necessarily be a surprise. It's rather simple really; the problem contains a contradiction.

Title: Re: Pop Quiz Riddle
Post by towr on Feb 21st, 2006, 3:28am

on 02/21/06 at 02:17:43, River Phoenix wrote:
 Basically, there are number of assumptions that are taken as axioms. Together, they lead to a contradiction. If we continue to take all of what is in the problem as axioms, then we conclude that the problem is simply nonsense.
It isn't nonsense. It's actually a common problem you face in dynamic epistemic logic with an update operator. Certain updates don't lead to knowledge because believing the information leads to a contradiction. This is exactly what the students are facing.

Title: Re: Pop Quiz Riddle
Post by Alky on Feb 21st, 2006, 9:16am
Hold on.

If the test is on tuesday, then they won't know the test is on friday thursday night. Only in the case where the test is on friday can they find out it is friday. If it is before friday then they obviously can't figure this out.

Maybe the problem we are facing is that we are assuming because we have "if the test is on friday then they will know it's on friday" is enough to show the test won't be on friday. If the test is on tuesday they won't see it coming. For example, to guess friday they would have to wait until thursday night (they only get one guess) so if it is any other day they lose.

Title: Re: Pop Quiz Riddle
Post by River Phoenix on Feb 21st, 2006, 5:10pm

on 02/21/06 at 03:28:04, towr wrote:
 It isn't nonsense. It's actually a common problem you face in dynamic epistemic logic with an update operator. Certain updates don't lead to knowledge because believing the information leads to a contradiction. This is exactly what the students are facing.

I was quoting some things that others had said earlier in the thread, namely that since you reach a contradiction, then one of the assumptions must have been wrong. Are you saying that this is not the case? I am very interested to understand how the problem works; I can see that it all boils down to the issue that "belief in X" --> "not X", as you say. What other problems contain this dilemma? Does it relate to Godel's incompleteness theorem?

Title: Re: Pop Quiz Riddle
Post by towr on Feb 22nd, 2006, 12:54am

on 02/21/06 at 17:10:20, River Phoenix wrote:
 I was quoting some things that others had said earlier in the thread, namely that since you reach a contradiction, then one of the assumptions must have been wrong.
The important things is, is that the contradiction is in your knowledge/beliefs about the world, not in the state of the world you're in. Propositional logic doesn't cut it in this case, what you need is some form of modal logic. So instead of just stating what is the case, you can talk about what you believe is the case (which isn't necessarily the same thing). And next to that you need something to 'update' your knowledge/beliefs.

It now all depends on how you define these modal operators; typically if you define 'knowledge', K(X) ("I know that X"), then whatever you know must be true: K(X) -> X. With belief that's not necessarily the case, you might belief things that aren't the case. Further you know what you know, and know what you don't know: K(X) -> K(K(X)) and ~K(X) -> K(~K(X))
If you remember Rumsfeld's famous words, it makes sense when you write it out.
there are A,B,C such that K(K(A)) & K(~K(B)) & ~K(~K(C))
There are known knowns, known unknowns and unknown unknowns.

Anyway.. What causes the paradox lies in the other aspect, updating your knowledge. Allowing every update to be succesfull may cause your beliefs/knowledge to become inconsistent. So one solution is not to allow such update to succeed.
It's believing what the professor says that causes the problem, not whether what he says is true.

http://www.illc.uva.nl/Publications/Dissertations/DS-2003-01.text.pdf goes into dynamic epistemic logic in section 4.3, and probabilistic epistemic logic in 6 (where there's a very brief mention of unsuccesfull updates on page 104)

Quote:
 Are you saying that this is not the case? I am very interested to understand how the problem works; I can see that it all boils down to the issue that "belief in X" --> "not X", as you say.
More accurately it's "belief in X" --> "belief in not X"
Reality doesn't change because you belief something, but your belief may become inconsistent if you'll believe anything people say.
And since you don't want inconsistent beliefs, you shouldn't belief everything; not even if it's true because it may become false by believing it.

Quote:
 What other problems contain this dilemma?
I don't know more examples from the top of my head, at least not ones that are basicly the same. There's a version with a quiz, with 5 boxes one of which has a prive (i.e. just like 5 days, one of which has a surprise test). And there's the hangman's paradox (make the execution of the prisoner as much of a surprise as possible.)

Quote:
 Does it relate to Godel's incompleteness theorem?
I don't think so. It's only a problem if you define your logic in the wrong way; that's to say, 'wrong' if you think it's bad to belief contradictions. I'm not sure how consistent people's beliefs are in reality though, so it may not have to be a problem to belief contradictory things

Title: Re: Pop Quiz Riddle
Post by River Phoenix on Mar 6th, 2006, 2:43pm

on 02/22/06 at 00:54:52, towr wrote:
 I don't think so. It's only a problem if you define your logic in the wrong way; that's to say, 'wrong' if you think it's bad to belief contradictions. I'm not sure how consistent people's beliefs are in reality though, so it may not have to be a problem to belief contradictory things

Descartes argues "I think, therefore I am", by using the construction that A="I think", and "disbelief in A" -> A
Does this mean that his argument might be wrong, and it is simply his act of doubting that is fallacious?

Title: Re: Pop Quiz Riddle
Post by towr on Mar 7th, 2006, 1:02am

on 03/06/06 at 14:43:40, River Phoenix wrote:
 Descartes argues "I think, therefore I am", by using the construction that A="I think", and "disbelief in A" -> ADoes this mean that his argument might be wrong, and it is simply his act of doubting that is fallacious?
I don't really see why there'd be a problem there. I mean, aside from that he might not be the one thinking.

What you have is:
Ex.T(x)
T(~Ex.T(x)) => Ey.T(y)
The implication is true. And the former follows from experience, he's aware that he thinks.

If he thinks that he thinks, than disbelieving that he thinks would be inconsistent, so logically he should believe he does think.
Whether that translates to knowledge is really more complicated. It's a bit odd to start with defining thought in such a way that you have to exist to do it; then go on to claim you think; and then to claim that therefor you exist.
For one thing, are you really doing what you defined as thought?

Title: Re: Pop Quiz Riddle
Post by shadebreeze on Jun 15th, 2006, 10:47am
My opinion (just an opinion, I am not good with riddles and I do not claim it's worth to be called a solution):

If the definition of "surprise test" is "a test so that NOW you can not determine which day it is", then it can be on any day with 1/5 probability. Obviously this can not be the case, it would be too easy.

Let's focus on the problem definition.

Let's say now that the definition is "a test so that at any day before it you can not determine which day it is". Then, since the whole domain is discrete, there is no "intermediate" point at which you can say "there has been no test today", because after a day there is immediately the next day, and on Thursday you can not assume the test has not been given on the same Thursday until it is already Friday, thus nullifying the students' logic.

If the definition is "a test so that at any time before it you can not determine which day it is", then the problem definition is probably incorrect, or at least has little sense.
Time is continuous, while days are discrete. This causes the paradox, because the students can reason in the continuous domain while the test can only be placed on points in a discrete domain. Maybe that makes sense apparently, but I am inclined to consider it a poorly specified (and thus potentially flawed) problem.

Now, what about a totally continuous space (so that the test can happen in any instant in the continuum and the definition of "surprise" is the continuous one)? Would the students' logic be consistent in a really continuous domain?
Definitely not.
For any two given instants, there are uncountable infinite instants between them, unless they are the same instant. Therefore, you could say that the test can not take place on the last instant of your time interval, but you can not define the "previous" instant, and you cannot "propagate" the "can not happen" backwards.

I hope I explained my point clearly, even though I doubt it (sorry I am not english born and I struggle to find words sometimes).

A side note that might be interesting...

I think this pop quiz relates to a paradox in probability laws as well.

The students say that on Thursday evening they know that the test will be on Friday if the prof didn't give it on any other day.
Therefore we can say that the conditioned probability of the test being on Friday is 1. By "conditioned" I mean "assuming the statement that the test has not been on the previous days is true". However, that statement itself has a probability.

At the beginning, the probability of the test being given in any single day is 1/5 (one divided by the number of valid days).
On Monday evening, if the test has not been given, the probability it will happen on any other day is 1/4. However, the probability that the test has not been given on Monday is 4/5. Therefore, the probability of the test being given on any other day than Monday is still 1/5 (because it's equal to 1/4 * 4/5).

You can iterate for all the days - the probability stays the same. But shouldn't it change day by day?

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Jun 15th, 2006, 2:27pm
The riddle still works if you discretise everything: suppose the test would be given at noon (in the regular slot for the lesson) and guesses have to be made at 4pm each day (at the end of classes) so the week becomes 10 discrete and separate chunks of time (4pm the previous Friday, noon and 4pm Monday, noon and 4pm Tuesday, noon and 4pm Wednesday, noon and 4pm Thursday, and noon Friday).

Odd numbered time blocks are for guessing when the test will be; even numbered for finding out if you were right (or having the test)

If you start with the assumption that each day is equally likely at 1/5, then it's hardly surprising if valid manipulations bring you back to the conclusion that each day is equally likely at 1/5.

Title: Re: Pop Quiz Riddle
Post by chanklas on Jun 20th, 2006, 9:17pm
Sorry if this duplicates someone else's post (I jumped from page 2 to page 9 on the forum, and at the end of page 9 it didn't look like someone posted this)...

...anyway, isn't the solution just that the professor knows the students are following their logic, which leads them to conclude that there can be no surprise test, so therefore when he DOES give one on Tuesday (or any day for that matter) it's sure to be a surprise?

Title: Re: Pop Quiz Riddle
Post by Icarus on Jun 22nd, 2006, 5:19pm
Yes, this is why the students are surprised. But the whole point of this riddle is, why did the students' logic fail?

They had an apparently logical proof that there could be no test, because it would not be a surprise. Yet, they had a test and were surprised by it. So, what was wrong with their proof?

One other thing: You can't be count on reading the latest discussion to let you know anything about what has been mentioned before. If there was one continuous discussion in this thread with everybody cognizant of the whole topic, then you might expect to get an idea of how it went by this page alone. But then you would have just spoiled it for the next guy because you have commented without knowing what went on before, effectively introducing a new thread of conversation that isn't dependent on earlier replies. So the next guy is going to be misled.

I say "would have" because, of course, you are not the first to skip. The conversation has been very disjointed, and so the only way to know what has been brought up is to read through it.

Title: Re: Pop Quiz Riddle
Post by Roy on Jul 25th, 2006, 12:19am
If the whole point of the riddle was, why was the students logic flawed? the answer would be they were wrong. that's it. i belive the question posed was, what was the flaw in the students logic.

Title: Re: Pop Quiz Riddle
Post by Icarus on Jul 25th, 2006, 1:21pm
The answer to why has multiple layers. Demanding that the answer be only the first layer, and the second layer is somehow an entirely different question (one which I also stated explicitly in my previous post), is ludicrous.

Title: Re: Pop Quiz Riddle
Post by Citizen on Aug 1st, 2006, 10:57am
Having read some very interesing positions in this thread, here's my two cents:

- With respect to the definition of surprise and the argument that you can't announce an upcoming "surprise" quiz.  I believe this argument is false - there are many courses where, at the beginning of the term, the students learn that there will be surprise quizzes during the term.  Empirically speaking, when they happen, they still tend to be surprising.  (Even though the student's logic could actually be applied in a situation where multiple surprise quizzes are expected)
- The professor, in combining the announcement with the offer has put the students into a paradox where a surprise test is unavoidable.  With them confident each day that the test cannot be that day, the professor can surprise them as he pleases (since they're not expecting a test).  This includes Friday, as, since they would enter class on Friday confident that the test would have been predictable that day, they won't be expecting it and therefore be surprised.
- The students failed to predict the day it would be on - they said it couldn't be on any day.  Thus he did not have to cancel the surprise quiz.
- Who's to say that the surprise has anything to do with the timing?  Perhaps the subject matter was the surprise.

Title: Re: Pop Quiz Riddle
Post by Roy on Aug 1st, 2006, 2:21pm
True, he may have before the class, just rolled a die to decide what day he would make the quiz, possibly he also made six on the die an option to not have the quiz as the student predicted. But he probably rerolled, i mean, where's the fun in no quiz?!

Title: Re: Pop Quiz Riddle
Post by Emmit1977 on Nov 20th, 2006, 12:24am
I think the answer may be as simple as next week never comes and the professor opts to give the quiz on tuesday of the CURRENT week.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Nov 20th, 2006, 8:20pm

on 11/20/06 at 00:24:46, Emmit1977 wrote:
 I think the answer may be as simple as next week never comes and the professor opts to give the quiz on tuesday of the CURRENT week.

The question of the riddle is not "how could the professor give a test?" but "what is wrong with the students logic that concludes that the test can't happen?" - the students' logic wouldn't be affected if the professor had said instead: "there will be a surprise test at one of these 5 times unless you manage to correctly predict which of these occasions the test would have been on"

Title: Re: Pop Quiz Riddle
Post by divisionst06 on May 23rd, 2007, 12:16pm
It's a paradox. The error to their logic is waiting until thursday to reply to the test. If the test is Thursday they are a day late with their response. The teacher makes the statement "GNice to see your thinking about it." Unless that's a varient in the riddle that would imply that more than one guess could be made. Thus, the error in their thinking would be trying to eliminate choices that are all valid rather than optimize their chances of getting the right answer.

Title: Re: Pop Quiz Riddle
Post by shadebreeze on May 29th, 2007, 9:47am
I think I got it.
I got it. :o
But I may be wrong, so please feel free to attack my solution from any angle. ::)

The professor says: "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." (format by me)

The use of the undeterminative "a" means that nothing prevents the professor from scheduling two surprise quizzes for next week (in fact the professor can schedule any number of quizzes he wishes, because they are surprise quizzes, so he is not obliged to tell the students).
If the professor can give two quizzes, the students' logic fails, because the assumption "if the quiz is on day X it can not be on any day after X" fails.
Note that the students have the possibility of canceling one quiz. However, they can not know which one of the two quizzes is the one whose date they have to guess in order to cancel it.
So they can not say "if the quiz was on Friday we would know by Thursday", because they can never know if the test given on, say, Tuesday, is the one added by the professor or the one they can actually cancel.

And it's all down to first-order logic. When I started writing the axioms of the problem, I realized that existance does not imply uniqueness.  :)

And now, here's a more mathematical explanation:

--- Problem description ---

Value set:
T = {1,2,3,4,5}

If the students guess, then there can be no test on that day:
ForAll t in T: (ScheduledTest(t) AND StudentsSay(t) ) => NOT GivenTest(t)

If the students do not guess right, the test is given:
ForAll t1, t2 in T: ( (t1 != t2) AND StudentsSay(t1) AND ScheduledTest(t2) ) => GivenTest(t2)
In other words, if a quiz is scheduled on any day different from what the students said, that quiz will be given.

The students can guess only one day (this is expressed using a mathematical trick with equivalence to shorten the expression):
ForAll t1, t2 in T: ( StudentsSay(t1) AND StudentsSay(t2) ) => t1 == t2

The professor schedules a test next week:
Exists t in T: ScheduledTest(t)     <---- this is the crucial one. There could be more than one t.

There is a test on Tuesday:
GivenTest(2)

--- End of problem description ---

The students' assumption is based on another, simpler assumption:
ForAll t1 in T: ScheduledTest(t1) => ( ForAll t2 in T: (t2 > t1) => NOT ScheduledTest(t2) )

Which is clearly and evidently false.
If he wanted, the professor could say
ForAll t in T : ScheduledTest(t)
Which does not contradict any of the problem's axioms.

You may say that's cheating.
I think it's perfect first-order logic.  ;D

Title: Re: Pop Quiz Riddle
Post by shadebreeze on May 29th, 2007, 10:38am
(Sorry, I may have skipped a passage which is not that easy)

The students' full assumption is "the test can't be on the last available day", and if you apply recursion you get to their conclusion.
Mathematically:

ForAll t1 in T : ( ( ForAll t2 in T : (t2 > t1) => NOT ScheduledTest(t2) ) AND ( ForAll t3 in T : (t3 < t1) => NOT GivenTest(t3) ) ) => NOT ScheduledTest(t1)

(for every t1, if no test is scheduled on any t2 greater than t1, and no test had been given on a t3 before t1, then no test can be scheduled on t1)

This works only if there is only one test scheduled for the week.
In that case, the students start from
SheduledTest(5)    <--- the test is on the last day
StudentsSay(5)     <--- it is not a surprise
And they have NOT GivenTest(1) AND NOT GivenTest(2) AND NOT GivenTest(3) AND NOT GivenTest(4) because there is only one test.
So both parts of the premise in the students' hypothesis are true, and they get a contradiction ( NOT ScheduledTest(5) ),  so ScheduledTest(5), which was their starting point, can't be true. And you can then use recursion.

It is immediately proven false when there are two scheduled tests or more and they are on different days, because the students can cancel only one of them. So, when they assume ScheduledTest(5) and StudentsSay(5) to start the recursion, there is at least another t so that GivenTest(t < 5).
So their assumption does not hold, and it is perfectly possible that ScheduledTest(5). The starting point of the recursion is removed, and the students' reasoning fails.

Here is an example:

ScheduledTest(2)
ScheduledTest(5)
StudentsSay(5)

Since the students can only say one day, the obvious conclusion is:
GivenTest(2)

Therefore,
... ForAll t3 in T : (t3 < t1) => NOT GivenTest(t3) ...
is false for t1 = 5, and they do not have the start of recursion.

Title: Re: Pop Quiz Riddle
Post by towr on May 29th, 2007, 10:54am
The students reasoning is wrong even if there is just one test.
Nothing fundamentally changes if the professor said "There will be exactly one surprise test next week, unless you can deduce on which day I plan to give it, in which case there won't be any test"

Title: Re: Pop Quiz Riddle
Post by Archae on May 29th, 2007, 11:34am
As many people have already said, this paradox is not clear on its terms: specifically, the notion of surprise.  Also, the fact that the teacher says that the test will be a surprise most likely alters the probability of the test being on day x.  For instance, if the teacher said that there would be a test the following week (without mentioning 'surprise'), distribution over the days would be more or less uniform.  However, once the teacher says that the test next week will be a surprise, the distribution changes for each day, assuming the teacher plans out which day the test will be on.  The teacher, though, could still roll a die to determine which day the test would be on, or any like action (picking a day out of a hat, etc.)
This information is not privy to the students or us.  So yes, they are correct in their logic that if they make it through Thrusday with no test, then the test will not be a surprise.  This logic shows that the students assume that the test can be on Friday.  Then the students dismiss Friday as a possibility, showing that they assume that the teacher picked the day of the test to be a surprise.  However, if the teacher had rolled a die or anything similar to choose the day of the test, the students' logic fails, and they cannot eliminate Friday as a possibility.
I think the paradox comes from these assumptions, coupled with an assumed tacit response from the teacher.  For if the students could honestly eliminate Friday as a possibility (and have their teacher agree that the test would not be on Friday), then their logic would work; eliminating the last day possible until there would be no days left.

Title: Re: Pop Quiz Riddle
Post by shadebreeze on May 29th, 2007, 2:02pm
Uhm, so it's not that there is more than one planned quiz... interesting.

Yes the definition of surprise is difficult to express even mathematically...
My definition was:
ForAll t1, t2 in T: ( (t1 != t2) AND StudentsSay(t1) AND ScheduledTest(t2) ) => GivenTest(t2)
but since the students' reasoning works perfectly fine in first-order logic if you take the assumptions I wrote, I am inclined to think there is a flaw in those assumptions, and the definition of surprise seems to be the weakest. So I am planning to consider a more "literal" one, and see what comes out of it.

Furthermore, there is another assumption I have made, which is that if a test is given, it had been scheduled.
But that is not necessary, it suffices that the professor just gives the test one day.

In fact, if we define a surprise test this way:
ForAll t in T : ( ( NOT StudentsBelieveTestIsOn(t) ) AND GivenTest(t) ) <=> SurpriseTest(t)

Then, we could say that the students' logic builds a system where

ForAll t in T : NOT StudentsBelieveTestIsOn(t)

So the professor has all the freedom to give the test any day.
Basically, the students' logic can not alter reality, it only alters their belief (as it has been suggested already in this thread). But if the definition of surprise is based on belief, then the problem may be easier than we think.

I'll write down all the mathematical dribble and see what I can get out of it, but I think I may be on a good lead.
Otherwise, if I cannot use that definition, writing down the maths is just an excercise in logic... I'll do it anyway because it is fun, but I won't post it if somebody tells me now that that definition of surprise is wrong.

Title: Re: Pop Quiz Riddle
Post by divisionst06 on May 30th, 2007, 5:06am
Correct me if I'm wrong:

Surprise acts as a modifier of the word "test" not the date of the test. If the test is a surprise, by definition it could just as easily mean the materials the test covers have not been established, the materials on the test would be the surprise in this instance. So, if the date of the test is not the surprise then the error in their logic is that the date of the test is a "surprise." It's the difference between "There is a surprise test next week," and "There will be a test on a surprise date next week."

Title: Re: Pop Quiz Riddle
Post by towr on May 30th, 2007, 6:08am

on 05/30/07 at 05:06:12, divisionst06 wrote:
 Correct me if I'm wrong:Surprise acts as a modifier of the word "test" not the date of the test. If the test is a surprise, by definition it could just as easily mean the materials the test covers have not been established, the materials on the test would be the surprise in this instance. So, if the date of the test is not the surprise then the error in their logic is that the date of the test is a "surprise." It's the difference between "There is a surprise test next week," and "There will be a test on a surprise date next week."
It's an interesting interpretation; and the students would still not be able to deduce when the test is.
But in common vernacular "surprise test" does mean that the time is unknown, not that the content is known. A math teacher that gives a surprise test wouldn't given a test on english literature; no matter how surprising that would be.

Title: Re: Pop Quiz Riddle
Post by divisionst06 on May 30th, 2007, 6:36am
I just came across that exploring multiple literal definitions of the word "surprise." It can mean to happen without warning or to happen without previously being established. Also, is there any way (provided we assign a value to each day of the week) in which tuesday would be the counter-example to their logic?

Title: Re: Pop Quiz Riddle
Post by towr on May 30th, 2007, 6:42am
I'm not sure what you mean. If there is any test at all (tuesday or otherwise), it's a countermodel to their logic, because that told them there wouldn't be a test.

Title: Re: Pop Quiz Riddle
Post by divisionst06 on May 30th, 2007, 6:47am
THIS MAY BE IT:

Looking more at the definition of surprise:

"a coming upon unexpectedly; detecting in the act; taking unawares."

The only way to defeat the quiz (so to speak) and to eliminate the indefinite "surprise" from the equation is to BE PREPARED.

If the students prepare and are ready for the quiz to be any day than the quiz is no longer a "surprise quiz" but rather just a "quiz."

The teacher is simply telling them to be ready for a quiz. If they are ready for a quiz, there is no quiz. The error in their logic has been listed several times differently and no doubt there are different errors to their logic. The only answer without a counter example is be ready for the quiz everyday making it no longer a surprise.

Title: Re: Pop Quiz Riddle
Post by divisionst06 on May 30th, 2007, 6:54am

on 05/30/07 at 06:42:24, towr wrote:
 I'm not sure what you mean. If there is any test at all (tuesday or otherwise), it's a countermodel to their logic, because that told them there wouldn't be a test.

I was merely speculating that there could be a mathematical equation in which Tuesday was the sole counter-example to the students logic.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on May 31st, 2007, 5:39pm
The sharpest form of the paradox comes when the professor says:

"There will be a test tomorrow if and only if you don't expect one"

The bright student reasons thusly: "I am not expecting there to be a test tomorrow, therefore, by his statement there will in fact be a test. However, I now expect there to be a test tomorrow, therefore there won't be a test..."

Title: Re: Pop Quiz Riddle
Post by srn347 on Aug 28th, 2007, 12:12pm
It can't be friday because by thursday you'll know. It can be thursday though because by wednsday if you think it's thursday, you could be wrong. You know on thursday it's on friday, but you'll have already used your guess.

Title: Re: Pop Quiz Riddle
Post by mahesh on Oct 27th, 2007, 10:15am

Since there was a contradiction the assumption that "there will definitely be a test this week and it will be a surprise" was wrong. The assumption led to the conclusion that there can be no test that week. Since the conclusion contradicts the assumption, the assumption has to be wrong.

If the assumption was relaxed to either "there will definitely be a test this week" or "there may be a surprise test this week" then it wouldn't lead to any contradictions.

The teacher went with the latter assumption and so should have the students. The teacher's logic would have been to pick any day at random and if the test day happened to be a Friday then he would simply not hold the test that week. If it happened to be on any other day he would proceed as planned.

Simple.

Title: Re: Pop Quiz Riddle
Post by Hippo on Oct 27th, 2007, 10:24am
Friday is OK as well.

Title: Re: Pop Quiz Riddle
Post by spazdor on Nov 7th, 2007, 5:25am
The professor's statement resembles part of a Godel sentence. Since the students are expected to be good, epistemically skeptical sorts, he is asserting "X is true iff there is no proof(in the students' decision schema) of X."

So indirectly, he's asserting that the students' decision schema is either incomplete or inconsistent.

Title: Re: Pop Quiz Riddle
Post by Farmer John on Jun 30th, 2008, 2:27pm
If we assume the professor and students to both be completely rational the problem breaks down in an interesting way which I think points to the heart of the paradox:

That a test cannot be a complete surprise unless it was predicted with absolute certainty to be on another day.  If both the professor and the students use the same assumptions and logic processes it is impossible for the professor to completely surprise the students as they would at least be aware of the possibility of the actual day under the assumed rule set.

So the only way to surprise the students is to rely on them making an incorrect assumption or to intentionally mislead / lie so as to confuse them into believing that they had predicted the test.  Interestingly all a logic professor would have to do to achieve this effect would be to lead them to the assumption that they actually could predict the day of the test.

If we assume that the students were mislead into predicting that the test would be on Monday, then we have to wonder what they were thinking on Monday night for it to remain a complete surprise.

Title: Re: Pop Quiz Riddle
Post by DMR on Nov 13th, 2008, 2:57pm
Everyone is over complicating this.

The question as stated is "What's the flaw in the students' thinking?"

The flaw was in the conclusion that the quiz couldn't happen.  They should have come to the conclusion that their reasoning contradicted a stated fact "We're going to have a surprise quiz next week".  In doing so they made an illogical conclusion.  They should have concluded that they didn't have enough information and that the exact day can't be figured out.   They never even considered (from the text of the riddle) that they had a conflict and thus couldn't resolve the answer with the given information.

All of the what if's go beyond the original question "Where was the flaw".  Once that Flaw was made the rest of any further discussion is moot.  If you mess up step 2 of 200 the last 198 don't matter.

Title: Re: Pop Quiz Riddle
Post by sippan on Dec 11th, 2008, 9:58am
Now I don't have time to page through this entire thread, so maybe this has been said before...

In my humble opinion, if you're following a logic chain that ends up contradicting the whole premise of the problem, then the chain is irrelevant to the case and invalid.

Title: Re: Pop Quiz Riddle
Post by Mickey1 on Oct 20th, 2009, 7:31am
(I apologize if somebody said this before)

It seems to me that
- the student's first reasoning is ok
- the second part "So it can't be on ANY day, so there's no quiz next week!" is in error.

The professor promised to cancel the quiz if the students could find out which day it was,
"if you can figure out what day it will be on, I'll cancel the quiz".

The student failed to do that, (they did something else, pointing out that the professors statement was meaningless) and therefore deserved to have the quiz.

Title: Re: Pop Quiz Riddle
Post by ThinkerzBlok on Jun 7th, 2010, 8:07am
The biggest flaw is that the quiz CAN be on Friday. No matter what happens the quiz can only be on one day and the students have a 1/5 chance of guessing it.

It is true that the students will know that the quiz is on Friday after Thursday is over, but that's because the students already gambled on the fact that the test was on Friday.

It is very possible that the students would use up their guess before then.

Anyone get what i'm saying?

Title: Re: Pop Quiz Riddle
Post by Grimbal on Jun 8th, 2010, 2:20am
Good point.

The professor's remark that the test might be canceled actually removes the paradox.

The professor can select Friday.  If the students say it is on Friday the test is canceled.  If they select another day, they won't know on Friday if the test was scheduled on the other day and was canceled or if it was scheduled on Friday.

Title: Re: Pop Quiz Riddle
Post by towr on Jun 8th, 2010, 3:17am
You can easily enough rephrase the problem in such a way that the paradox remains.
The problem is that being told a true fact can lead to inconsistent knowledge if you were to accept it as true.

Title: Re: Pop Quiz Riddle
Post by RrjH on Jun 9th, 2010, 4:39pm
This is actually a riddle based off of an old paradox, the paradox was similar, but it was an execution instead of a quiz, and a prisoner instead of students.

The idea is that since the students figured out that it wouldn't be on Friday, and therefore couldn't be on Thursday, and Wedensday, and so on, that by giving them the quiz on one of those days, they would not see it coming.

In a sense it was their own logic that surprised them when he ave out the test.

There is no math to this riddle, ignore the people that say there should be. It's a plain and simple paradox with one answer.

Title: Re: Pop Quiz Riddle
Post by Mickey1 on Jun 10th, 2010, 12:48pm
Let’s have another look at the teacher’s statement:
“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."

There is no contradiction here – the teacher could have said:
“We're going to have a surprise quiz next week. You will not know what day it is...”. This is what I originally and erroneously read into the statement. These two latter statements are contradictory, but not the teacher’s.

“I am not telling you” does not amount to a statement that can be used to deduce anything at all. Therefore the students cannot avoid the quiz based on anything the teacher has said. The teacher’s promise is worthless in terms of avoiding the quiz but valuable in misleading the students and myself to the Friday to Monday induction.

A related issue: if stated as my alternative above - or as  rmsgrey - would the students' Friday-to-Monday reasoning be logically valid. I think it would.

Title: Re: Pop Quiz Riddle
Post by rmsgrey on Jun 11th, 2010, 7:14am
An interesting variation on the puzzle is the one where the students, by cramming the night before a test, can significantly improve their marks in the test - but if the test isn't the next day, the effect is lost (though they can always have another cramming session).

With this variation, it becomes clear that the students can successfully improve their grades by "predicting" the test by cramming for up to five nights (they can stop once the test happens) - or they can fail to improve their grades by not cramming at all. In the former case, their successful prediction relies on them being allowed to take as many guesses as they need, which is altogether unsurprising...

Title: Re: Pop Quiz Riddle
Post by Bobzor on Apr 28th, 2012, 1:30pm
Because of the student's certainty of the tests incapability of being a surprise having the test at any day would be a surprise even when they have somehow deduced the date of the test.