# Is this a paradox? Or what? **URL:** \ **Category:** Factual Questions\ **Created:** [July 11, 2000, 9:12am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194 "2000-07-11T09:12:35Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![shike](https://avatars.discourse-cdn.com/v4/letter/s/977dab/32.png) [@shike](https://boards.straightdope.com/u/shike)\ **Post date:** [July 11, 2000, 9:12am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/1 "2000-07-11T09:12:35Z")
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 ---
**Author:** ![ricepad](https://avatars.discourse-cdn.com/v4/letter/r/b38774/32.png) [@ricepad](https://boards.straightdope.com/u/ricepad)\ **Post date:** [July 11, 2000, 9:18am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/2 "2000-07-11T09:18:42Z")
Once you’ve concluded that it won’t be given, it’ll be Monday. ---
**Author:** ![Alessan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/alessan/32/465_2.png) [@Alessan](https://boards.straightdope.com/u/Alessan)\ **Post date:** [July 11, 2000, 10:05am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/3 "2000-07-11T10:05:30Z")
Simple answer: 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 **ever** know which day it is you are going to get it - is a lie. In two weeks, you’ll know. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [July 11, 2000, 5:46pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/4 "2000-07-11T17:46:38Z")
If you want your head to spin even more, there’s a good page of paradoxes [here](http://www.geocities.com/CapitolHill/Lobby/3022/). 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. ---
**Author:** ![Dragwyr](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dragwyr/32/2855_2.png) [@Dragwyr](https://boards.straightdope.com/u/Dragwyr)\ **Post date:** [July 11, 2000, 6:23pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/5 "2000-07-11T18:23:18Z")
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. ---
**Author:** ![CheapBastid](https://avatars.discourse-cdn.com/v4/letter/c/8baadc/32.png) [@CheapBastid](https://boards.straightdope.com/u/CheapBastid)\ **Post date:** [July 11, 2000, 6:36pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/6 "2000-07-11T18:36:44Z")
[The Unexpected Hanging](http://www.greylabyrinth.com/Puzzles/puzzle051.htm) ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [July 11, 2000, 7:36pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/7 "2000-07-11T19:36:15Z")
The teacher should be teaching math instead of making stupid riddles. Yes, that’s my final answer. ---
**Author:** ![Sofa\_King](https://avatars.discourse-cdn.com/v4/letter/s/58956e/32.png) [@Sofa\_King](https://boards.straightdope.com/u/Sofa_King)\ **Post date:** [July 11, 2000, 8:18pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/8 "2000-07-11T20:18:34Z")
Ah, but that’s the beauty of the thing, sailor. The test is on Cantor’s Infinite Set theory, and the teacher is Zeno. ---
**Author:** ![IzzyR](https://avatars.discourse-cdn.com/v4/letter/i/2acd7d/32.png) [@IzzyR](https://boards.straightdope.com/u/IzzyR)\ **Post date:** [July 11, 2000, 8:57pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/9 "2000-07-11T20:57:55Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 11, 2000, 9:11pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/10 "2000-07-11T21:11:35Z")
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. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [July 11, 2000, 9:11pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/11 "2000-07-11T21:11:55Z")
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… ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 11, 2000, 9:18pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/12 "2000-07-11T21:18:53Z")
> [@](#): > > \*Originally posted by IzzyR \* > **This “paradox” resolves itself.** The paradox of the self-resolving paradox. How does it do that? > [@](#): > > \*\*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. \*\* And what if Wednesday passes? Since you claim it can’t be given Friday… Ah, now I see. ---
**Author:** ![IzzyR](https://avatars.discourse-cdn.com/v4/letter/i/2acd7d/32.png) [@IzzyR](https://boards.straightdope.com/u/IzzyR)\ **Post date:** [July 11, 2000, 10:12pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/13 "2000-07-11T22:12:55Z")
Slowly, this time. 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.) ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 11, 2000, 10:31pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/14 "2000-07-11T22:31:55Z")
> [@](#): > > \*Originally posted by IzzyR \* > **Slowly, this time.** That wouldn’t have been necessary if it had just resolved itself like it’s supposed to. ---
**Author:** ![Diceman](https://avatars.discourse-cdn.com/v4/letter/d/22d042/32.png) [@Diceman](https://boards.straightdope.com/u/Diceman)\ **Post date:** [July 11, 2000, 11:54pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/15 "2000-07-11T23:54:43Z")
I love The Unexpected Hanging. I first ran into it at [http://www.greylabyrinth.com](http://www.greylabyrinth.com). They have a bunch of really good logic puzzles. ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [July 12, 2000, 1:38am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/16 "2000-07-12T01:38:30Z")
Somewhat coincidentally, this paradox, along with its explanation, was covered in last month’s _Discovery_ magazine. Or maybe _Scientific American. Discovery_, though, I think. ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [July 12, 2000, 8:23am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/17 "2000-07-12T08:23:05Z")
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. ---
**Author:** ![hawthorne](https://avatars.discourse-cdn.com/v4/letter/h/c89c15/32.png) [@hawthorne](https://boards.straightdope.com/u/hawthorne)\ **Post date:** [July 12, 2000, 2:51pm UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/18 "2000-07-12T14:51:22Z")
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. picmr ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [July 13, 2000, 5:51am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/19 "2000-07-13T05:51:35Z")
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. ---
**Author:** ![voltaire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voltaire/32/313_2.png) [@voltaire](https://boards.straightdope.com/u/voltaire)\ **Post date:** [July 13, 2000, 6:59am UTC](https://boards.straightdope.com/t/is-this-a-paradox-or-what/25194/20 "2000-07-13T06:59:13Z")
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…