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Nicodemus

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Surprise Quiz - My Thoughts [Oct. 19th, 2005|08:43 am]
Nicodemus
[This entry refers to the logic problem in my previous entry.]

A lot of good comments; it's clear people were really thinking about it! This is a classic problem that shows up in different guises and is still debated.

First, some preliminaries. In the problem, I tried to define "surprise" as not knowing which day the quiz was. I should have added "at any time prior to the quiz day" just to be clear. Given the problem's particular definition, saying that there will be a "surprise" quiz sometime next week is not contradictory.

I'm also going to work from the assumption that the professor didn't intentionally lie to them. One might assume the opposite, but once you assume that the problem itself contains lies, you might as well assume that the problem didn't exist. (If you can demonstrate that there's an inherent contradiction, that's different.) Plus, the professor's original statements (there was a quiz and it's timing was a surprise) came true.

I'd like to consider two weaknesses within the problem.
(This was correctly identified by kuddlepup, traveller_blues, twopiearr, and bogglerat, at least.)

Flaw in the Students' "Proof"

Time is a slippery thing to combine with formal logic. The students' reasoning confused future knowledge with present knowledge. If we expand the students' logic premises, it becomes a bit more obvious:

IF it is Thursday night AND the quiz hasn't happened THEN the quiz occurs on Friday.
(Since this relies on it being Thursday, we could predict on Thursday a Friday quiz. Thus the quiz cannot be on Friday, as they concluded. But, we can only know this on Thursday.)

IF it is Wednesday night AND the quiz hasn't happened AND the quiz cannot be on Friday THEN the quiz occurs on Thursday.
(The students concluded that all of the atecedant conditions here are true. Assume the first two, for the sake of argument. But the third, that Friday is ruled out, came from our previous logical step. During that step, we assumed that it was now Thursday night; in this step, we assumed that it's now Wednesday night!)

So the logic of the students (as I presented it) is undone by some minor but contradictory assumptions.



(I think that ristin, pippinbear, tafyrn, and xolo were all heading in this direction. Some good explanations, too!)

True Statements May Be Unprovable

This problem is entangled with Godel's Incompleteness Theorem, as are most self-referential logic puzzles (e.g. "This sentence is false."). Basically, it's possible to have a logically consistent collection of statements yet be unable to prove some things within that system.

The students' logic was misapplied. They assumed that they could deduce which day the quiz was, thereby negating the surprise. Nowhere did the problem say that the quiz could be predicted.

In fact, the problem itself stated that the quiz would be a surprise, thus ruling out this avenue of thought! This is where some people (mistakenly, I think) come to the conclusion that the professor lied: If the students could deduce which day the quiz was, then the professor's assertion that it would be a surprise is false. But I think they cannot.

Can we demonstrate that the students cannot predict the quiz?

Let's take these statements:
(A) There is a quiz
(B) The quiz is/will be a surprise
(C) The day of the quiz is known
(D) It is possible to deduce the day of the quiz

The problem basically gives us the following axioms:
A ("there will be a surprise quiz next week")
B ("there will be a surprise quiz next week")
B -> !C ("It will be a 'surprise' because they won't know on which day he will give it.")
Note that it doesn't tell us about D.

Based on definitions, we can also say:
D -> C ("IF it is possible to deduce the day of the quiz THEN the day of the quiz is known." We're assuming these are good logic students. :)

Now, using this information:
1. Given B
2. Given B -> !C
3. Thus !C
4. Given D -> C
5. Thus !D

That is, since we know statement C is false, D must likewise be false. (Otherwise, the rule D -> C would be contradictory.)

Thus it is not true that it is possible to deduce the day of the quiz. The students were overly optimistic.


Anyhow, that's my analysis of this one. Please feel free to post alternate approaches, gaps in my reasoning, etc. And thanks for the great comments on the puzzle! (Yay! I've got all these clever readers! :)
LinkReply

Comments:
[User Picture]From: traveller_blues
2005-10-19 05:50 pm (UTC)
At this rate, you're going to get a character cameo in my mystery novel for NaNo.

*grins*

"Puzzles. Puzzles help me think. Briana, what's the puzzle here?"

-Traveller
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[User Picture]From: nicodemusrat
2005-10-19 08:11 pm (UTC)
I strive to be interesting, as always. Glad to inspire. :>
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[User Picture]From: polrbear
2005-10-19 07:36 pm (UTC)
I think their error is in thinking that logic can solve every problem. In this case it is not really a logic problem, it is a simple issue of probability. On Monday there is a 1 in 5 chance of the quiz, Tuesday is 1 in 4, and so forth till Friday where it is 1 in 1 or 100% probable. The actual day is still unpredictable, or !D, since they can only know the probability of each day at the end of the previous day.
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[User Picture]From: nicodemusrat
2005-10-19 08:10 pm (UTC)
Correct. And we can use logic to show that they cannot use logic to predict the test. Fun stuff. :)
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[User Picture]From: seinnichts
2005-10-19 08:23 pm (UTC)
Logic cannot solve every problem. And in cases of low probability of occurrence, you might need to rely on Intelligent Design.



When the creator of the quiz wishes to keep the quiz a secret, naturally he needs to have it placed on the day of greatest improbability. Given that Monday is the most obvious non obvious day, given the 20% chance of occurrence, most students should expect a high probability of quiz here, even with the low random chance probability. Tuesday, the next day, with a 25% of quiz is naturally the first non obvious day for the quiz. Since the lack of quiz on Monday means that it might indeed be placed on a random day, the expectation for the quiz should be less on Tuesday than on Monday.

And indeed, the quiz was on Tuesday.

It all makes sense once you consider that an intelligent mind was behind when the quiz would randomly appear.
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[User Picture]From: seinnichts
2005-10-19 08:24 pm (UTC)
.. and when you construct the justification for the answer after you know the answer :)
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[User Picture]From: foobart
2005-10-20 07:11 am (UTC)

I call bogosity...

...on your part 2 premises. Or at least, I think it's overly complicated.

IN short, what you deduced was this: The problem says the quiz is a surprise. Since the problem says it's a surprise, the student's can't possibly have deduced the day of the quiz.

One might just as easily give a parallel problem:

A professor told the students "Today, I will be adding 2 + 2 and informing the class of my results.". Adding 2 + 2 is impossible. What's the problem in the story?

and then follow this chain of logic.

  1. The problem says adding 2 + 2 is impossible
  2. Therefore the professor was overly optimistic

I think your first take on the problem is more accurate. Here's another way to look at it:

The professor says it will be a surprise. We're interested in whether or not the professor is mistaken.

At the time the professor makes the statement, clearly it can be a suprise; assuming a constant distribution, there is a 1/5 chance for each day, so the students can't know which day it will occur.

If there is no quiz on Monday, then there is a 1/4 chance for each of the remaining days.

The last day, then, becomes unique. If there is no quiz on Thursday, there is a 1/1 chance of the quiz on Friday. So the students can't get through class on Thursday without knowing when the quiz is.

But this doesn't mean the quiz isn't a surprise -- it merely means the last possible moment for it to be a surprise is Thursday's class. This falls directly out of needing 2 options to have a surprise. Because the last day is unique; using the last day as the seed for an inductive proof is not valid.

However, if the professor had said "The quiz will be a surprise at the moment it is given", things get more interesting. Then it turns into a "This statement is false" kind of paradox.
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[User Picture]From: mrpuzuzu
2005-10-20 04:58 pm (UTC)
Heya, this is unrelated to the quiz (which was pretty cool).

I've been trying to email Kit_Ping, and I also left comments in her journal, but she hasn't responded. I lost your guys phone number, and have just been trying to figure out when I should drop by the spare key for while we're away (she's going to feed our poor kitty). I tend to be free during the day, so was thinking I could leave it in an envelope in your mailbox. I may have lost your address too, though (doh!) Shannon and I are also going to want to take the two of you out to eat afterwards to thank you for saving our poor starving cat.
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