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LobowolfXXX Inner circle La Famiglia 1196 Posts |
I imagine some of the people who hang out here have seen this, but it's an interesting one. A professor tells his class, "Next week, we will have a surprise quiz one day."
One of his students says, "Well, it obviously can't be on Friday, because if we haven't had it by the end of Thursday, we'll know it will be Friday, and it won't be a surprise. So it must be somewhere from Monday to Thursday." At which point another student said, "Well, if THAT'S the case, then it can't be Thursday, because if we haven't had it by the end of Wedenesday, we'll know it must be Thursday, Friday having been eliminated, and it won't be a surprise. So it must be somewhere from Monday to Wednesday." At which point another student said... And so on.
"Torture doesn't work" lol
Guess they forgot to tell Bill Buckley. "...as we reason and love, we are able to hope. And hope enables us to resist those things that would enslave us." |
TomasB Inner circle Sweden 1144 Posts |
This is one fantastic problem I have severe trouble getting my head around. Most people would agree that the quiz can't possibly be on the last day, since that could never be a surprise then. But once you agree with that, the rest of the reasoning seems to just take its own correct course.
Some people say that it's a semantic problem - because the word "surprise" is used, but I've seen formulations of this problem that gets away with that, such as: It is monday and a professor in logic tells his class that there will be a quiz during the week. If someone can tell him, the day of the test that "The quiz will be today." and put forward a logic reason for it, the test will be skipped. The professor really wants the students to take the quiz. Obviously he can't have the quiz on friday since everyone would come friday morning and say "The quiz will be today." Right? /Tomas |
TomasB Inner circle Sweden 1144 Posts |
In what I wrote above there needs to be a punishment. Let's say that a student that says "The quiz will be today." when it isn't, will fail the class. Otherwise they could ask that question every day if there is no risk.
/Tomas |
magicjohn2278 Special user Isle of Man UK 544 Posts |
I haven't seen that before, It's good!
Tomas's version works better for me, in the original you have to argue that there can be no surprise quiz "next week" on any day, (If we deduce that it's on Monday, then it is no longer a surprise, and therefore can't exist.) |
magicjohn2278 Special user Isle of Man UK 544 Posts |
... so eliminating the paradox...
A professor tells his class, "Some time this term, we will be having a surprise quiz." - When is the quiz? ... The quiz has to be NOW. How's that? |
Steve Martin Inner circle 1119 Posts |
I know this as the prisoner who is told he will be executed by the end of the week, but will not know when he wakes up that that day he will be executed.
It is one of my favourite puzzles. Although I have discussed it several times with people, I do not have an adequate explanation for the paradox.
Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Albert Einstein |
landmark Inner circle within a triangle 5194 Posts |
Maybe a related problem:
How can you make a hill of sand? I pose the following propositions: 1) One grain of sand is obviously not a hill. 2) Adding one more grain of sand to something that is not a hill, cannot turn it into a hill. Thus I can start with one grain of sand and keep adding a grain at a time. By propositions 1 and 2 at no time will I have a hill. Jack Shalom
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Josh the Superfluous Inner circle The man of 1881 Posts |
Don't I have enough to think about without reading this section!?!?? Now my ears will be smoking for the rest of the day. Stupid puzzles. Of course if he emailed the test outside of class time, it would be a surprise. Or he could pass out blanks sheets of paper on Thursday and tell the students to wait to flip them over. When they do he could yell "Surprise! the test is tomorrow."
Is Jack's puzzle related? I can't get my head around this one yet.
What do you want in a site? "Honesty, integrity and decency." -Mike Doogan
"I hate it, I hate my ironic lovechild. I didn't even have anything to do with it" Josh #2 |
dlhoyt Regular user 176 Posts |
The beauty of this problem is that when, say Wednesday, rolls around the professor can come into class and announce the quiz. And it will be a complete surprise!
Martin Gardner published a book, The Unexpected Hanging and other Mathematical Diversions, with a chapter devoted to this paradox. (He didn't solve it.) The book has a long bibliography of papers dealing with the paradox. Great fun reading it! |
TomasB Inner circle Sweden 1144 Posts |
Actually Gardner gives an explanation in his book "Gotcha" where the problem is called "The Unexpected Tiger".
Martin says that most people agree with the first step in the logic, that it can't possibly happen on the last day but goes on to say that even this first step is faulty. Let's say that we have already reasoned that the test can't happen on the very last. Suppose all days went by until friday morning. Can we still say that the test can't possibly be on friday as we have just reasoned it couldn't be? Martin writes: "No, because if he makes such a deduction, he might open the door and find an unexpected tiger! Indeed, the entire paradox holds even if only one room is involved." I can't say that I follow Gardner's reasoning completely. He ends with the following reasoning: The consensus among logicians is that although the professor knows he can keep his word, there is no way that the students can know it. Therefore there is no way they can make a valid deduction about the absence of a test on any day, including the last day. I still don't understand that reasoning, which is what makes this one of my favourite problems. Sureley the professor won't even consider putting the test on the very last day? Confused, /Tomas |
Nir Dahan Inner circle Munich, Germany 1390 Posts |
I never liked paradoxes...
but to me it is just a contradiction within the question, maybe I am too naive... nir |
TomasB Inner circle Sweden 1144 Posts |
Which question do you mean? Put yourself in the situation of the professor. Would you put the test on friday? I wouldn't, if I really wanted the students to take the test. Is there a flaw so far in that reasoning? According to Gardner (if I understand him correctly) it is.
I've accepted that I probably won't be able to understand why the reasoning is wrong. On the other hand, here could be an explanation: If you make some assumptions and reduce them in absurdum and find a contradiction you know that your assumtions were wrong. That's a common way of proving things. If we accept that friday can be eliminated, we know that he can't possibly put the test on thursday, since if the test hadn't been on wednesday everyone would know thursday morning that the test would be then. The same reasoning cuts out wednesday, tuseday AND, this is the important point, monday. So if our assumptions are correct he can't have the test on any day BUT we know that the test would be during the week, hence our reasoning is wrong. I can't say exactly what went wrong and that's pretty darn annoying. /Tomas |
Josh the Superfluous Inner circle The man of 1881 Posts |
I think I'm closing in on it. Once Friday is ruled out by logic, it becomes a candidate for a surprise. When Thursday ends with no test, the logicians would be surprised that it IS to be given on Friday. If this is true, Friday can be included with the other days and the surprise can happen on any day of the week.
I think this holds up as long as the "surprise" doesn't have to occur on the day of the test. Semantics: On the other hand, did the professor void any surprise by announcing a specific event that would occur within a time frame? If he said tomorrow we are having a test, but didn't specify A.M. or P.M. could he call it a surprise test? By making his statement he is saying "Next week we are having a test". Just because he calls it a surprise doesn't make it so. What if this were the three shell game with no trickery? If I told you the pea [the test] was under one of the shells [the days], you wouldn't necessarily be able to guess correctly which one. But, would you be surprised when it was revealed? I don't think so. I told you it was somewhere in there. Doesn't this apply?
What do you want in a site? "Honesty, integrity and decency." -Mike Doogan
"I hate it, I hate my ironic lovechild. I didn't even have anything to do with it" Josh #2 |
Steve Martin Inner circle 1119 Posts |
I like your idea, but unfortunately it doesn't apply to this paradox in the form of the prisoner who is to be hanged on a certain day before the end of the week, with the condition that when he wakes up each day, he won't know for sure that he will be hanged that day.
Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Albert Einstein |
Josh the Superfluous Inner circle The man of 1881 Posts |
We are discussing the test puzzle. Let me struggle with one paradox at a time. If I haven't figured the first puzzle out, I will not concede that it is the same as the prisoner puzzle (or the sand puzzle for that matter).
What do you want in a site? "Honesty, integrity and decency." -Mike Doogan
"I hate it, I hate my ironic lovechild. I didn't even have anything to do with it" Josh #2 |
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