I love a good paradox – especially ones I can’t see a resolution to. I’ve run into this one a few times and wanted to look into it further.
The story is as follows. A teacher is worried that her class isn’t working consistently through the term, choosing instead to “cram” on the night before an exam. So she announces to the class that there will be a quiz on the work they will cover this week. It will be a surprise quiz, sometime next week. The students, not knowing which day the quiz is on, will not have the option of staying up the night before. They must work consistently so that they are always prepared for the quiz.
One enterprising student, however, quickly realises that the test cannot be on Friday. His reasoning is sound – suppose that a student hadn’t worked consistently through the week. Suppose it is Thursday night, and the quiz hasn’t happened yet. Then the student knows that they should stay up all night and cram, which defeats the purpose of the surprise quiz. Thus, the quiz won’t happen on Friday.
So far, so good. But suppose that it is Wednesday night, and the quiz hasn’t happened yet. The quiz must be on Thursday or Friday. But we just saw that it won’t be on Friday. Thus it must be on Thursday. So the student should stay up all night cramming. Which defeats the purpose of the surprise quiz. Thus, the quiz won’t happen on Thursday.
You can see guess what happens next. On Tuesday night, the student would know that the quiz was on Wednesday. On Monday night, the student would know the quiz was on Tuesday. Thus, the student knows that the exam is on Monday, so it is obviously not a surprise quiz. The student concludes that there is no such thing as a surprise quiz.
On Tuesday, the teacher hands out the surprise quiz. The student is, frankly, surprised.
Any mathematicians out there will recognise an induction. Mathematical induction is a form of proof that works like this:
We are attempting to prove a set of statements U(n), where n = 1,2,3 … For example, U(n) could be the mathematical formula:
U(n): 1 + 2 + 3 + … + n = (n2 + n)/2
Since we have an infinite number of statements, we can’t check them one-by-one. Instead, we line them up like dominoes, and then push the first one over. More precisely, we prove the following
Step 1: If U(n) is true for any particular value of n (say, n = k), then it is true for the next value of n (i.e. n = k + 1).
Step 2: U(n) is true for n = 1 i.e. U(1) is true.
Step 2 says U(1) is true. Thus, by Step 1, U(2) must be true, because U(1) “knocks it over”. But then, by Step 1, U(3) must be true, because U(2) knocks it over. And so on.
Returning to the surprise quiz paradox, a number of the discussions of the paradox on the net (e.g. here and here ) suggest that any attempt to put the paradox in the form of an induction will fail because the term “surprise” cannot be given a precise, mathematical meaning.
However, such a formulation has been given here, in one of the comments. The problem is set out as follows:
Premise 1: On exactly one day out of the next n days, there will be a quiz.
Premise 2 (the “surprise” requirement): On the evening of day k (given that the quiz didn’t happen on days 1,2,…, k), there does not exist a proof that the quiz will be on day k+1.
The claim, then, is that these premises are inconsistent. This is shown as follows.
If the quiz was to occur on day n, then on the evening of day n – 1, there would be a simple proof that the quiz would occur on day n:
Proof:
- The quiz must occur on one of the days 1,2,3 … n (premise 1)
- The quiz did not occur on days 1,2,3 … n-1 (by assumption)
- Thus, the quiz must occur on day n.
Since premise 2 forbids such a proof, the quiz cannot occur on day n.
Now the induction shows itself. If premise 1 holds, but premise 2 requires that the quiz cannot happen on day n, then we can formulate a new premise:
Premise 1a: On exactly one day out of the next n – 1 days, there will be a quiz.
But Premise 1a, along with premise 2, can be used to formulate this premise:
Premise 1b: On exactly one day out of the next n – 2 days, there will be a quiz.
And so on. Thus, we reach the conclusion that the announcement of a surprise quiz to the class is self-defeating.
(At this point I am reminded of a passage in John Barrow’s book, “Impossibility”, where he presents a similar argument along the lines that you can only predict the future if you keep the prediction to yourself. That is a topic for another post.)
At this point, I will not attempt to resolve the paradox, because I’m not sure what the resolution is. I’m told that many philosophers have written papers on this very paradox in its various forms (other versions include a man sentenced to hang). I’ll simply leave you with a list of references to further investigation, and invite you to provide your solutions.
Wikipedia: Unexpected hanging paradox
Stanford Encyclopedia of Philosophy: Epistemic Paradoxes
PrawfsBlawg – especially the discussion in the comments.
Cornell course notes: Discrete Mathematics in Computer Science – Induction
“What is a proof?” – Robin Cockett, University of Calgary
Letters to Nature 
I find the surprise conclusion amusing: that after conducting this reasoning to show that the quiz can’t happen, the class is then surprised to find out that it actually does. So they are surprised. 🙂
Surprise is intimately related to probability. I will have a go at modelling this situation using Bayesian probability theory – but not now, it’s 2 am.
Actually, I don’t think you’ve made your second premise precise enough. It says “… there does not exist a proof that …”, but you don’t specify what proof system we’re allowed to use. If you restrict us to first-order logic, for example, then I think premise 2 is a second-order axiom, so it can’t be used in any such proof.
I don’t know enough about higher-order logics to say much more, but I suspect you’ll have difficulty getting the second premise to be fair game for the proof system that it specifies. Perhaps some sort of Gödel numbering system can get you out of it, but then you might end up with it being undecidable whether the quiz can be on Thursday. Undecidable, that is, from within the logic that you specify, but not necessarily from outside it.
I haven’t actually read much of what you linked to, unfortunately, so I don’t know if I’m covering ground that you already know about. Perhaps I’ll read more on a day when I’m not recovering from New Year’s Eve.
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It seems to me that the student was right. What the teacher meant to say was that the quiz could surprise you on days Monday thru Thursday, or we might just have it on Friday.
He had to acknowledge that on the last day for the test, if it hasn’t been given, it won’t be a real shocker.
It’s interesting, of course it’s all rubbish anyway. By surprise exam we mean that the day of the test is not disclosed in advance. No predictions about any of the students’ psychological states upon being receiving the quiz are actually made.
I don’t think that changing the definition of “surprise” helps. I’d contend that the definition given in the post is close to the best one: something is a surprise if you don’t know it’s coming. A test on thursday is a surprise test if, on wednesday night, you don’t know whether or not there will be a test tomorrow.
Your intuition seems to be right, though. A test on monday to thursday would be a surprise. Well, maybe not thursday.
I agree. Changing what the student means by surprise quiz just avoids the problem. What I did, was say that he is right. It is “paradoxical” to say that something can be a surprise and expected at the same time. Impossible to have a surprise quiz on the only day left to have it.
But so what? “Everyone” knows this. That when the professor announces a “surprise quiz” at all is interesting. Interesting in the way that makes me say, “Language is funny, isn’t it?” But that’s as far as I go.
What I see in this puzzle is a play on the word surprise, and the observation that things happen before their deadlines (properly), and what is the real mystery for me – not everyone agrees with me.
On second thought, perhaps the professor should describe the quiz as secret. No, that’s no better. Maybe he should just say, “quiz which may surprise you unless if and when there remains only one possible time for the quiz to be given.” That would be more precisely what he means, but does that make it better? Actually, it might surprise someone even after they have all the information they need. At least, that’s what they claim. …. I digress.
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I apologize, I’ve come around to the opinion that the professor’s use of the word surprise is best. I was half-serious about finding an improvement. Something less confusing. My mistake. I was confused.
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The problem is that, given the inductive logic, what she has said is self-contradictory; so of course when you assume what she says is true you wind up in paradox. Consider the simpler case: The professor says I’m giving you a surprise quiz tomorrow. Then you can, assuming what she says is true, reason that she can’t give you a test tomorrow, since if she did it wouldn’t be a surprise, and that she can because I now don’t expect one. The “paradox” is just that you can get contradictory conclusions by starting from contradictory axioms.
That is wrong. If you assume she will surprise you with a quiz tomorrow, you cannot reason that you will not be surprised. You cannot expect the unexpected.
You could expect a quiz tomorrow tho, be surprised that its not a surprise, and refuse to take it on the grounds that the Professor said nothing about having an expected quiz.
Wait… that doesn’t work either.
“But suppose that it is Wednesday night, and the quiz hasn’t happened yet. The quiz must be on Thursday or Friday. But we just saw that it won’t be on Friday.”
This is where the reasoning goes wrong. The previous information that it won’t be on Friday was gained under the assumption that it is Thursday, an assumption which does not apply if it is Wednesday.
Where do I collect my reward?
I know on Wednesday night that the quiz won’t be on Friday, reasoning thusly:
* It will be a surprise quiz.
* If the quiz is on Friday, then it won’t be a surprise.
* Thus, the quiz will not be on Friday.
I don’t need to assume that it is currently thursday evening to make the above argument. It holds at any time.
I agree that it can’t be Friday. However, for all the other days, the argument that it can’t be on that day rests on the assumption that, on the previous evening, it hasn’t happened yet. So the argument for Friday holds at all times, but not the corresponding arguments for the other days.
Say it’s Sunday. Can I now say “it won’t be Wednesday”? No.
There obviously CAN be a surprise.
Maybe this is the solution: The student concludes, using the reasoning outlined above, that it cannot happen on any day. So, whenever it happens, he is thus surprised. 🙂
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The problem is with the assumption that the test cannot happen on Friday, because that wouldn’t be a surprise. Actually, it would be a surprise, but it would be a surprise on Thursday when it was announced there would not be a test that day, meaning the test would indeed be on Friday, which nobody was expecting.
The solution is simple, recognize that your teacher has uttered a paradox. A paradox is what you have when you have a set of contradictory statements. There will be a surprise quiz this week implies there won’t be a surprise quiz this week. Since the statement contradicts itself, there is annihilation or explosion: you can conclude nothing, or everything, the quiz may be a surprise, might not, may not even happen at all.