In high school, ages ago, I remember discussing the following problem and never being satisfied with the solution. If someone could clarify the exact wording and source of the problem I'd appreciate it.
What I loosely remember is as follows. A teacher tells her students on a Friday that they are to have a test next week. However, she adds, they won't know the day before that they're going to have a test on the following day.
The students, being somewhat savvy, deduce that if it transpires that they don't get tested by the end of the following Thursday then they'd have to be tested on Friday, which would mean that they know this beforehand, so the test cannot possibly be on Friday. And so, inductively, cannot be on Thursday, and so on.
This "logic" is very unsatisfying, so I need to find out what the actual form and solution of the problem is.
Wow Peter, thanks for that. I like their analysis.
It reminded me of a great article I read years ago that dealt with the issue of self-referential paradox. In fact, at the time I tried applying the techniques therein towards the surprise examination, however I didn't get anywhere.
It's a paper by Noson Yanofsky entitled "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points" (http://arxiv.org/abs/math/0305282)
I think you'll enjoy reading it. I read the paper on a long train ride loved working out how given "Ann believes that Bob believes that Ann believes that Bob has a false belief about Ann" asking "does Ann believe that Bob has a false belief about Ann?" results in a paradox. This is supposedly called Brandenburger’s Epistemic Paradox.
Thanks, downloaded. I have to read the paper I quoted a few more times before I grok it fully. Perhaps it will be clearer after reading the one you cite, which appears most intriguing.
Thanks again!
EDIT: Just started reading. Mind already blown by the clarity of the "limitation, not paradox" idea. Slowing down for a longer, deeper, read.
Hm... that is indeed a weird problem. If you don't get tested on Monday, that is a surprise, because you expect to get tested on Monday. Then if you don't get tested on Tuesday, that is surprising, because you've inductively determined that Wednesday through Friday are ineligible. The rest of the days follow the same logic.
In effect if you just tell someone "you will have a surprise test next week", that is contradictory, since they can reasonably expect to have a test at any moment during the following week and the test can therefore never be surprising, just the absence of the test.
In high school, ages ago, I remember discussing the following problem and never being satisfied with the solution. If someone could clarify the exact wording and source of the problem I'd appreciate it.
What I loosely remember is as follows. A teacher tells her students on a Friday that they are to have a test next week. However, she adds, they won't know the day before that they're going to have a test on the following day.
The students, being somewhat savvy, deduce that if it transpires that they don't get tested by the end of the following Thursday then they'd have to be tested on Friday, which would mean that they know this beforehand, so the test cannot possibly be on Friday. And so, inductively, cannot be on Thursday, and so on.
This "logic" is very unsatisfying, so I need to find out what the actual form and solution of the problem is.
This has bugged me for decades.