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Dean Foster
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The definition of "know"  
« Reply #100 on: Jun 10th, 2003, 7:47am »
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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
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #101 on: Jun 10th, 2003, 8:08am »
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there is no way to predict the unpredictable..
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #102 on: Jul 7th, 2003, 6:36pm »
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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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #103 on: Jul 8th, 2003, 1:54am »
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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"
 
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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #104 on: Jul 8th, 2003, 7:35am »
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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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #105 on: Jul 8th, 2003, 8:36am »
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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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #106 on: Jul 8th, 2003, 5:21pm »
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on Jul 8th, 2003, 1:54am, 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... Wink
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #107 on: Jul 9th, 2003, 12:47am »
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on Jul 8th, 2003, 5:21pm, 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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #108 on: Jul 9th, 2003, 1:42am »
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I've tried to formalize the riddle, and here's what I think it is.

 
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.
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Re: anyone know the answer to the pop quiz riddle?  
« Reply #109 on: Jul 9th, 2003, 7:35am »
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The reason Tuesday's test came as a total surprise is because the Professor had also given a test on Monday.
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Re: Pop Quiz Riddle  
« Reply #110 on: Jan 5th, 2004, 12:54pm »
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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.
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Re: Pop Quiz Riddle  
« Reply #111 on: Jan 15th, 2004, 4:38am »
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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...
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Re: Pop Quiz Riddle  
« Reply #112 on: Jan 15th, 2004, 7:44am »
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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.
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Re: Pop Quiz Riddle  
« Reply #113 on: Jan 15th, 2004, 6:05pm »
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The riddle asks a question.
 
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.    
 
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Re: Pop Quiz Riddle  
« Reply #114 on: Jan 15th, 2004, 6:36pm »
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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
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Re: Pop Quiz Riddle  
« Reply #115 on: Jan 22nd, 2004, 7:57am »
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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
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Re: Pop Quiz Riddle  
« Reply #116 on: Apr 5th, 2004, 2:57am »
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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.
 
ThoughtsHuh
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Re: Pop Quiz Riddle  
« Reply #117 on: Apr 5th, 2004, 7:25am »
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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...
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Re: Pop Quiz Riddle  
« Reply #118 on: Apr 6th, 2004, 2:34am »
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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
 
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Re: Pop Quiz Riddle  
« Reply #119 on: Apr 6th, 2004, 11:11am »
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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...
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Re: Pop Quiz Riddle  
« Reply #120 on: Apr 7th, 2004, 4:33am »
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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!
 
 
 
 
 
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Re: Pop Quiz Riddle  
« Reply #121 on: Apr 7th, 2004, 5:51am »
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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]
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Re: Pop Quiz Riddle  
« Reply #122 on: Apr 8th, 2004, 1:23am »
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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
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Re: Pop Quiz Riddle  
« Reply #123 on: Apr 8th, 2004, 4:00am »
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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
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Re: Pop Quiz Riddle  
« Reply #124 on: Apr 9th, 2004, 5:47am »
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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]
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