A co-worker of mine gave me this little thing to solve, NO problem I thought at first. But the more
I think about it, the more it annoys me. OK no more talking here it is:
A teacher says to his class, “I’m going to give you a surprise test next week. And there is no way you can know which day
it is you are going to get it.”
Ok, simple enough. Since you mustn’t know the day before that the test is the next day (the teacher wants the students to
study in the weekend, not the night before the test) you can rule-out Friday. Because if the test hasn’t come when school
ends Thursday, they know it must be on Friday and can study that night. OK. But on Wednesday night, if the test hasn’t come
they know it must be on Thursday right? Ok, so the test must be on Wednesday, but then you know that on Tuesday, right??
and so on… and so on… I hope you get the picture… now can you explain how this could be so? When is the test?
My mind boggles, I hope you can save my sanity because it’s slipping awayyyyyyyy
The teacher’s riddle can mean two things:
a. There is no way you can know **right now **which day it is you are going to get it - this is quite true, but it no longer applies once you enter the zone of possibility.
b. There is no way you can everknow which day it is you are going to get it - is a lie. In two weeks, you’ll know.
If you want your head to spin even more, there’s a good page of paradoxes here. The unexpected exam is mentioned a little bit down the page.
Basically, the original reasoning that eliminates Friday is wrong, since one cannot know until arriving at class whether or not the test will be given.
panama jack
I think I have a mental illness, but my doctor says I’m just imagining it.
Martin Gardner explores this very paradox in his book “The Unexpected Hanging”. I’ve always like this one. It made me use my brain at a time when I probably would not have. It also has lots of other mathematical “diversions” as well.
This “paradox” resolves itself. Becouse the logic described by shike ultimately eliminates every single day of the week, it eliminates none. In other words, a student pondering at the beginning of the week which day it will be, cannot resolve the question based on the logic. Thus any “knowing” based on the logic is false. Thus the student truly does not know which day it will be given.
Obviously it can’t be given on Friday though. Once Thursday passes there is no logical paradox to “confuse” the student, so the student can indeed know which day it will be given.
Alessan has the right reasoning here… What the teacher means, is that right now you don’t know when it’ll be. She obviously doesn’t mean that you’ll never know when it’ll be… You’ll eventually find out, one way or another. Until you find out, it’s still a surprise.
Yes, IzzyR obviously it can’t be given on Friday. The latest it could possibly be given is Thursday, but once the students realize this, it can’t be given on Thursday, either…
At the point at which you stand on Wednesday night contemplating the situation, you know that it cannot be given on Friday. But you also know that it cannot be on Thursday. Therefore you are confused. Therefore you know nothing.
On Thursday night, there is only one day left. So you know that it will be given on Friday.
(Unless you allow for the possiblity that it will not be given at all as being one of the variables. In this case you are still confused on Thursday night, and still know nothing. But when I was in school, this was never a possibility.)
Somewhat coincidentally, this paradox, along with its explanation, was covered in last month’s Discovery magazine. Or maybe Scientific American. Discovery, though, I think.
OK, I’m home now, and I’ve checked the reference. It’s in the June 2000 issue of SciAm page 108.
The resolution printed there is paraphased as follows:
The paradox is logically equivalent to the students saying every morning “the test is today.” They expect it every day, and one day they happen to be right. The author of the article describes it as a cheat, something that looks like a parakox but really isn’t.
He also mentions that he’s had many arguments with mathematicians about this, so it’s not been definitively resolved, I guess.
The importance of the problem is the doubt it casts on using the method of backward induction to solve games.
Using backward induction, you must conclude that the test cannot be on the last day etc and the problem collapses, yet the test is a surprise when it comes.
Other game solutions rely on backward induction to reach a purported rational strategy. The paradox poses difficulties for this method of solution.
For example, in the finitely repeated prisoners’ dilemma, cooperation (usually) collapses under backward induction.
Some suspect that the flaw in backward induction as a form of reasoning lies in imagining that one is at the last node of the game under circumstances which somehow violate the rules of the game, meaning that conclusions drawn about the last node (and therfore the second last etc) are invalid.
This an example of Godel’s thereom at work: that is, there is a statement which is easily decided outside of a system, but because it refers to the system, it can not be decided within the system. Now, obviously the students can’t know what day the test is, because the only information they have is the teacher’s statements. If the students reach any conclusion about what day the test will be based upon these statements, they will have proved the statements false (after all, one of the statements was that they wouldn’t be able to figure out what day was the test, and if they do figure out what day the test is, then obviously this statement is false), and therefore their reasoning will be based on falsde statements, and therefore not valid. So we know that the students won’t know. But the students can’t know this. I’ve szeen this simplified to two days, but why not simplify it to one day? Suppose the principal comes into the room and asks whether there will be a test any time soon. The teacher replies “The students are unaware, and will remain unaware until tommorrow, of the fact that there will be a test tommorrow.” For the principal, this statement poresentgs no problem: it is clear that there will be a test tommorrow. But if the students believe the teacher’s statement that there will be a test tommorrow, why not also believe the statement they won’t know about it? And so for the students, there is a paradox.
But for practical intents and purposes, the student who needs to cram the day before the test would have to cram every day before Friday, if that’s when the test is given. So, in effect the professor achieved his goal by discouraging the practice of cramming. Even though the students will finally know on Thursday that the test is tomorrow, they didn’t know that at the beginning of the week.
Basicly, this is a quasi-paradox only if the students KNOW that they can’t EVER know when the test will be. And it would be pointless for the professor to tell them that. All he has to do to achieve the desired effect would be to say that the test would be sometime next week.
But then again, if it made sense it wouldn’t be a paradox…