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Topic: Pop Quiz Riddle (Read 65459 times) 

Gamer555
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Re: anyone know the answer to the pop quiz riddle?
« Reply #25 on: Jul 31^{st}, 2002, 5:50pm » 
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I got it! Be the undealer, 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).


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anshil
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Re: anyone know the answer to the pop quiz riddle?
« Reply #26 on: Jul 31^{st}, 2002, 11:43pm » 
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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.)


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aliks
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Re: anyone know the answer to the pop quiz riddle?
« Reply #27 on: Aug 1^{st}, 2002, 10:32am » 
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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)


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aliks
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Re: anyone know the answer to the pop quiz riddle?
« Reply #28 on: Aug 1^{st}, 2002, 10:33am » 
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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)


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Nicodemus
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Re: anyone know the answer to the pop quiz riddle?
« Reply #29 on: Aug 1^{st}, 2002, 12:35pm » 
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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 fancypants 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!)


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Jonathan the Red
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Re: anyone know the answer to the pop quiz riddle?
« Reply #30 on: Aug 1^{st}, 2002, 1:44pm » 
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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.


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icon
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Re: anyone know the answer to the pop quiz riddle?
« Reply #31 on: Aug 1^{st}, 2002, 2:15pm » 
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nice! not much else to say here but agreement


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Alex
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Re: anyone know the answer to the pop quiz riddle?
« Reply #32 on: Aug 2^{nd}, 2002, 9:59am » 
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I think everybody's overcomplicating it with their definitions of "surprise" and terms like "surprisealltheway." 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."


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Jonathan the Red
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Re: anyone know the answer to the pop quiz riddle?
« Reply #33 on: Aug 2^{nd}, 2002, 4:15pm » 
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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.


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Scott Schneider
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Re: anyone know the answer to the pop quiz riddle?
« Reply #34 on: Aug 6^{th}, 2002, 3:58am » 
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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.


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anshil
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Re: anyone know the answer to the pop quiz riddle?
« Reply #35 on: Aug 6^{th}, 2002, 4:27am » 
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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 surpisealltheway (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, surprisealltheway)  Of what other timelines can you think this situation could have gone through?

« Last Edit: Aug 6^{th}, 2002, 4:29am by anshil » 
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Ryan Lawrence
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Re: anyone know the answer to the pop quiz riddle?
« Reply #36 on: Aug 6^{th}, 2002, 11:40pm » 
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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.


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Steven Noble
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Re: anyone know the answer to the pop quiz riddle?
« Reply #37 on: Aug 8^{th}, 2002, 4:19am » 
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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 rereading 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 ~QT) Then we have the assertion "There is a surprise quiz" (or Just Q) Which leaves us with the following (~QT) & 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.


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horseisahorse
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Re: anyone know the answer to the pop quiz riddle?
« Reply #38 on: Aug 8^{th}, 2002, 9:15am » 
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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.


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anshil
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Re: anyone know the answer to the pop quiz riddle?
« Reply #39 on: Aug 8^{th}, 2002, 9:40am » 
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Steven can you tell us about the 2 prisoners?


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Gerard
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Re: anyone know the answer to the pop quiz riddle?
« Reply #40 on: Aug 8^{th}, 2002, 1:47pm » 
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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 nonacademic. 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.


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anshil
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Re: anyone know the answer to the pop quiz riddle?
« Reply #41 on: Aug 8^{th}, 2002, 10:06pm » 
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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?


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Gerard
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Re: anyone know the answer to the pop quiz riddle?
« Reply #42 on: Aug 9^{th}, 2002, 12:24pm » 
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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.


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ilia Denotkine
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Re: anyone know the answer to the pop quiz riddle?
« Reply #43 on: Aug 9^{th}, 2002, 1:39pm » 
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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.


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Ryan Lawrence
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Re: anyone know the answer to the pop quiz riddle?
« Reply #44 on: Aug 9^{th}, 2002, 4:35pm » 
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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 Catch22 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 facevalue the answer I provided earlier makes sense, because the recursive logic relies on the premise of overanalyzing "surprise."


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bling0
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Re: anyone know the answer to the pop quiz riddle?
« Reply #45 on: Aug 10^{th}, 2002, 1:14am » 
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Ok, this was a question in a CS textbook, but I don't remember which one. I do remember, however, that this was under the induction section. Note that the student's logic is as follows: Base case: IF (a) it is Thursday AND there has been no test, THEN we know that the test cannot be on Friday. Inductive step: Given that we know that the test cannot be on Friday, then we can apply induction to the same thing for Thursday. Conclusion: there can be no test Ryan said it earlier I think. The student's *logic* is completely faulty because the student's base case does not hold. The logic relies on the fact that IF it is Thursday AND there has been no test. What if it is Thursday AND there has been a test? The logic doesn't hold.


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George Wright
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Re: anyone know the answer to the pop quiz riddle?
« Reply #46 on: Aug 10^{th}, 2002, 6:07pm » 
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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...n1. 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. Rereading 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.


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Steven Noble
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Re: anyone know the answer to the pop quiz riddle?
« Reply #47 on: Aug 10^{th}, 2002, 7:22pm » 
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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.


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Archon
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Re: anyone know the answer to the pop quiz riddle?
« Reply #48 on: Aug 15^{th}, 2002, 11:07pm » 
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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 selfcontradiction. 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.

« Last Edit: Aug 15^{th}, 2002, 11:16pm by Archon » 
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anshil
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Re: anyone know the answer to the pop quiz riddle?
« Reply #49 on: Aug 15^{th}, 2002, 11:52pm » 
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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.


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