### The Surprise Exam, and More Surprises

If you’re the sort of person who reads this blog, you’re likely to be familiar with the paradox of the unexpected hanging, which has been floating around since 1943 but achieved popular notoriety around 1969 through the writing of Martin Gardner. But you’re less likely to be aware that the unexpected hanging plays a central role in a wonderful new piece of serious mathematics related to algorithmic complexity, Godel’s theorems, and the gap between truth and provability.

The unexpected hanging might as well be a surprise examination, and that’s the form in which I present this paradox to my students every year: In a class that meets every weekday morning, the professor announces that there will be an exam one day next week, but that students won’t know exactly which day until the exams are handed out.

The students, of course, immediately start trying to guess the day of the exam. One student (call him Bob) observes that the quiz can’t be on Friday — because if it is, the students will know that by Thursday afternoon. After all, if Monday, Tuesday, Wednesday and Thursday mornings have all passed by, only Friday remains. A Friday exam can’t be a surprise exam.

A more thoughtful student (call her Carol) observes that this means the quiz must be on one of Monday, Tuesday, Wednesday or Thursday — and that if it’s on Thursday, they’ll know that by Wednesday night. After all, Friday’s ruled out, so if Monday, Tuesday and Wednesday have passed by, then only Thursday remains. That rules out a surprise exam on Thursday.

Another student (call him Ted) observes that thanks to Bob and Carol, we know the exam must be on one of the first three days of the week — which means that if it’s not on Monday or Tuesday, it must be on Wednesday. Therefore if it’s on Wednesday, they’ll know this by Tuesday night. Scratch Wednesday from the list of possibilities.

Now Ted’s girlfriend Alice points out that the exam can’t be on Tuesday either. Whereupon Bob concludes that the exam must be on Monday. But wait a minute! Carol points out that if they know the exam will be on Monday, it can’t be a surprise. Therefore no surprise exam is possible.

The students, relieved, decide not to study. But they’re awfully surprised when they show up in class the following Tuesday and the professor hands out an exam.

Where did the students go wrong? There is no consensus among the many philosophers and logicians who have considered this problem. The great Willard Van Orman Quine believed that Bob went wrong at the very beginning when he ruled out Friday. (According to Quine, Bob’s argument fails to distinguish between a proof that the exam can’t be on Friday and a proof that the students will know that the exam can’t be on Friday.) Other deep thinkers have accepted Bob’s argument (agreeing that the exam can’t be on Friday) but refused to accept Carol’s (thus refusing to rule out Thursday). You can, if you wish, read a pretty comprehensive survey of this literature here. But even among those who think Bob (or Carol) is mistaken, there is little agreement about exactly why they are mistaken.

Now, I happen to think the surprise examination paradox is pretty interesting as a pure intellectual exercise. But it’s also got important applications. I use it in the classroom to illuminate our discussion of the underlying “backward induction” technique, which economists (and especially game theorists) use all the time in serious arguments. Much more recently, the surprise examination has been used to illuminate some key concepts in mathematical logic, which I alluded to back in the first paragraph of this post. That’s the coolest part of all, and I’ll tell you all about it later in the week.

#### 59 Responses to “The Surprise Exam, and More Surprises”

1. 1 1 Mike H

The students’ reasoning is only valid if the professor’s statement is correct. Since their (correct) reasoning leads to a contradiction, the professor’s statement is not correct. Therefore their reasoning is invalid, and none of the days can be ruled out.

2. 2 2 JLA

This is neat; I’m looking forward to the follow up.

Backwards induction solutions are powerful, but they sometimes seem to be supported by weird off the equilibrium path beliefs. I’m troubled by solutions in the centipede game http://en.wikipedia.org/wiki/Centipede_game_(game_theory), but I can’t for the life of me formalize exactly why. It just seems so bizarre to expect players to play on path after an off path deviation.

3. 3 3 JLA

This is roughly the idea I had in mind:

Say player one plays off the equilibrium path. According to the backwards induction solution, player two still plays on path, because he believes that player one is Bayesian rational and will play on path in the future. But if player one was Bayesian rational, he would never have played off path in the first place. So player one can’t be Bayesian rational. But then player two shouldn’t necessarily believe that player one will play on path in the future.

4. 4 4 J Storrs Hall

Agree with Mike H. If the professor had said the students wouldn’t know “when” the exam would be instead of “what day”, leaving them to guess an instant out of a continuum, the argument wouldn’t work and the paradox disappears.

5. 5 5 Alan Gunn

Mike H’s argument needs one more line: “And therefore the professor’s statement was correct.”

Because the students were surprised, so what he said was true. That’s what makes the problem fun.

6. 6 6 Harold

I like Mike H’s analysis – it seems the Prof got it wrong, and he was making a claim he couldn’t support.

But what if we look at is another way. The prof says there will be an exam next week – probability 1. The fact of the exam cannot be questioned. He also says it will be a surprise – the students will not know which day until the papers are handed out, also with a probability of 1 – or that the probability of them knowing which day is 0.

If we say Prof assigns the day randomly – probability of each day = 0.2. This is decided before the week starts. Obviously, once the day is chosen, the actual probability for the chosen day is 1, and for the others it is 0. Lets say Friday is pickled – i.e. probability that the exam will be on Friday = 1

The students knowledge is different. On Monday, they can only assume a P=0.2 for each day. By Tuesday, that had become 0.25 for each remaining day, assuming the test was not on Monday. By Thursday, this has become P=1 for Friday. However, our original statement was that the probability of the students knowing the day was 0 – yet here we have P=1. Therefore both statements cannot be true.

The problem arises, as J Storrs Hall says, because the question chops up the days into 5 discrete chunks, but he says the student swill not know until “the papers are handed out” – a sub-division of the chunks. If he had said the students would not know which day until the day of the exam, he could spring a surprise – assuming he could set the test up to midnight on Thursday. There would alweays be a 0.5 probability of the exam occuring on Thursday, right up to the moment if became Friday.

7. 7 7 Jonathan Campbell

The teacher could solve the problem by saying there is a 99% chance of a pop quiz next week, and a 1% chance of a quiz on Friday.

8. 8 8 Steve Landsburg

Harold, J Storrs Hall, Jonathan Campbell and others: I agree that if the teacher had said something different, there would be no problem. If he’d said that the exam would be at a random *time* rather than a random *day* there’d be no problem. But this is no more relevant than observing that if he’d said “We’re having a (non-surprise) exam on Thursday”, there’d be no problem. The fact that there would be no problem in some *other* circumstance does not resolve the problem in *this* circumstance. And the reason we have a problem in *this* circumstance is because, as Alan Gunn points out, the promised surprise is in fact delivered.

9. 9 9 Edmond

My view is that there is a misunderstanding of the situation.

Ruling out the last day of the set is perfectly valid, but where the students run into trouble is continuing this same elimination repeatedly. Friday is out (after class thursday the test being friday is certain), thus you could consider eliminating Thursday as well, but that may be one assumption too far. Not a very eloquent way of explaining this, but perhaps my meaning will come through.

How rigidly logical or infallible do the students believe their professor to be? If they do not believe the professor to be infallible (I know, I know, but we are talking hypotheticals here), only Friday can be eliminated (after class on Thursday would make a Friday exam certain), but even on Thursday there remains a possibility that the professor is fallible and has made an error by giving it on Friday.

Bonus solution – the professor will give the exam whenever he is finished writing it, thus not even the professor knows when that will occur.

10. 10 10 Harold

There is a re-formulation in the link:
“Exactly one of five students, Art, Bob, Carl, Don, and Eric, is to be given an exam.
The teacher lines them up alphabetically so that each student can see the backs of the students ahead of him in alphabetical order but not the students after him. The students are shown four silver stars and one gold star. Then one star is secretly put on the back of each student. The teacher announces that the gold star is on the back of the student who must take the exam, and that that student will be surprised in the sense that he will not know he has been designated until they break formation. The students argue that this is impossible; Eric cannot be designated because if he were he would see four silver stars and would know that he was designated. The rest of the argument proceeds in the familiar way”.

It is simpler to think of just 2 students, Alice and Bob.

Does this work? They work out that if Bob is designated, then he will not be surprised, therefore it cannot be him. If Bob is ruled out, then Alice is the only option, so it cannot be her either. Having ruled out both, whichever it is is bound to be surprised when they find out.

It may help to clarify to see what happens if the prof. did not say that anyone would be surprised. Then if it was on Bob, he would see Alice with the silver star, and not be surprised it was him. If it was on Alice, she would be surprised. Now what if the prof did say they would be surprised. Bob would then see the silver star on Alice, and would know that it must be him. He would also know that it could not be him, from his earlier reasoning. Paradox. This gaurantees his surprise when he finds that he does indeed have the gold star on him. He is only surprised if he takes account of what the Prof. said.

11. 11 11 thomasblair

>>I’ll tell you all about it later in the week.

I suppose we can conclude that it won’t be Friday…

12. 12 12 Jonathan Campbell

“Therefore no surprise exam is possible.

The students, relieved, decide not to study.”

The first sentence should read “therefore it is not *guaranteed* that there will be a surprise exam.” The 2nd sentence would not follow from this modified 1st sentence.

I don’t understand why this is a paradox. I agree that the teacher cannot truthfully guarantee that he/she will give a surprising quiz next week. Surely the students should be willing to estimate probabilities of a quiz on any day the following week (e.g. 20% for each). If they do that, they will find, assuming there are at least 2 days with nonzero probabilities, that there may, or may not, be a surprising quiz next week (Steve, you conveniently chose Tuesday in your example. If you’d chosen Friday, there would in fact not have been a surprising quiz). Any statement which contradicts this finding will be wrong, not paradoxical.

13. 13 13 Ken B

@Jonathan Campbell: You are not seeing the argument. It’s not about an unmentioned guarantee. The argument is this. It can’t be the last opportunity (per Bob). So by induction (per Carol, Ted, Alice, and Paul Mazursky) it cannot be any opportunity. The question is, why is that wrong.

14. 14 14 Jonathan Campbell

Ken B: I perfectly well understand the sentences you’ve written. The error is that Bob can only say that a surprise quiz cannot be given on Fri. He cannot say that a quiz cannot be given on Fri. Given that, the induction fails.

15. 15 15 Harold

Jonathan Campbell- try it with a 2 day weekend. It cannot be Sunday, because I would not be surprised since once Saturday was over, I would know it must be Sunday. That means it must be Saturday – therefore I will not be surprised when it is Saturday. Therefore it cannot be Saturday. Therefore it cannot be either Saturday or Sunday, therefore whichever day it is, I will be surprised. I think the same paradox remains.

Maybe it still works for a single day. I tell you you will have an exam next Tuesday, and you will be surprised about it. You conclude that you cannot have an exam, since it is no longer a surprise. hence you are surprised when you get one.

16. 16 16 Ken B

What makes a surprise? Is it that you cannot predict with better than a random chance of being right? Today the students can bet on one of MTWTF. They have it seems a 1/5 chance of being right. Does that make the situation a surprise? I toss a fair coin. Is the result a surprise?

So what if there is only one choice, and a random guess is 100%. Is that a surprise?

17. 17 17 Jonathan Campbell

Harold: you, like Ken B, are not distinguishing between exams and surprising exams. I never suggested that I disagree with the claim that it is impossible to guarantee that a surprise exam will be given over the course of a 5 day (or 2 day) period. A claim to the contrary (such as the teacher’s) is a false claim, not a paradoxical claim.

18. 18 18 Jonathan Campbell

Ken B: An exam is a surprise if and only if the students are not 100% certain, the night before the exam, that the surprise will be the following day.

19. 19 19 Ken B

@Jonathan Campbell: you are assuming I am endorsing a particular argument and I have not done so. In fact an alert reader might surmise from my ‘surprise’ questions that I am doing just what you –sans evidence — assume I am not.

Your remarks seem contradictory and confused. In one you remark that the teacher is just wrong to claim he can give a surprise exam [" A claim to the contrary (such as the teacher’s) is a false claim" -- but the teacher claimed he can do so, and did in fact do so.] Yet in another you claim that he can — since the chances will be less than 100%.

20. 20 20 ThomasBayes

Surprise = reduction of uncertainty.

If the professor ‘randomly’ picks the day for the exam, then she will give her students log2(5)=2.32 bits of surprise if she tells them the exam day in advance, say on the Sunday before that week. If she tells them on the day of the exam, then she will still give them 2.32 bits of surprise, she’ll just give it to them a little at a time.

Suppose, for example, the exam is on Wednesday. When the students learn they don’t have an exam on Monday, then they will get 0.32 bits of surprise. When they learn they don’t have an exam on Tuesday, they will get 0.42 bits of surprise. When they learn they do have an exam on Wednesday they will get 1.58 bits of surprise. Thursday and Friday will bring no surprise. The total surprise, then, is 2.32 bits.

If the exam comes on Friday they will get 0.32 bits on Monday, 0.42 bits on Tuesday, 0.58 bits on Wednesday, 1.00 bits on Thursday, and 0 bits on Friday. Again, the total surprise is 2.32 bits.

If maximum surprise is your goal, then tell them in advance. Otherwise, you’ll be spreading the surprise over several days. It’ll still be the same total surprise, though.

There is no way to have any bits of uncertainty left on Friday, so the professor is promising to do something that can’t be done. Why is that a paradox?

21. 21 21 Jonathan Campbell

A claim to the contrary of the claim “it is impossible to guarantee that a surprise exam will be given over the course of a 5 day (or 2 day) period” is false, as I said.

22. 22 22 Ken B

@ThomasBayes: In the problem at hand, does “it will surprise you” have a clear meaning that differs from “your best prediction will be one derived from the maximum entropy distribution” — which here must be the uniform distribution? So in a 2 day sitiaution it will be a surprise if you cannot beat a 50-50 guess. If that is right then it looks like you CAN have a one day surprise — you cannot beat the 1 day 100% uniform prediction. We get a degenerate definition of surprise this way.

If you don’t mean something precise like that as a defintion of surprise then you have prevarication.

23. 23 23 Ken B

@Jonathan Campbell: Amidst a welter of double negatives you endorse the “no surprise is possible” argument. You wrote:
A claim to the contrary of the claim “it is impossible to guarantee that a surprise exam will be given over the course of a 5 day (or 2 day) period” is false

Strip off the double negatives and you get it is impossible to guarantee that a surprise exam will be given over the course of a 5 day (or 2 day) period

You also wrote An exam is a surprise if and only if the students are not 100% certain, the night before the exam, that the surprise will be the following day.

Since the students cannot predict the date this is a contradiction — a surprising exam IS possible by this definition.

24. 24 24 ThomasBayes

@Ken B: “It will surprise you” means you will have less uncertainty after you learn what I have to tell you. “It will not surprise you” means I’m not telling you anything you didn’t already know so your uncertainty remains the same. My precise definition to quantify the amount of ‘surprise’ is Shannon’s information entropy.

On Sunday night you think the exam is equally likely to be on any day of the week so your uncertainty can be quantified as log2(5)=2.32 bits. It is in this sense that I say you will receive 2.32 bits of surprise if you learn the exam day before Monday morning.

Instead, suppose you show up on Monday morning and learn one of two messages: a) the exam is today; or b) the exam is later in the week. If you learn that the exam is today, then your uncertainty drops from 2.32 bits to 0 bits, and you get a big surprise. If you learn the exam is later in the week, then your uncertainty drops to log2(4)=2 bits, and you get a smaller (0.32 bits) surprise. On Monday, then, you either get the full 2.32 bits of surprise, or you get 0.32 bits of surprise. Either way, you won’t be able to get more than 2 bits of surprise on Tuesday.

Regarding the 2-day situation, if the professor tells you the exam is either Monday or Tuesday, then you have log2(2)=1 bit of surprise. It makes no difference if you learn the day in advance or learn it in class on Monday. Either way, you’ll get the 1 bit of surprise all at once.

I think the phrase ‘knowing exactly’ is a confusing thing. Knowing exactly the outcome of a coin toss with 1/16 probability of heads is different than knowing exactly the outcome of a coin toss with 1/2 probability of heads. The first case will give, on average, about 0.33 bits of surprise: log2(16)=4 bits of surprise if it is heads; log2(16/15)=0.09 bits of surprise if it is tails. The second case will give log2(2)=1 bit of surprise either way.

25. 25 25 Jonathan Campbell

Ken B: You are still confused. You say “Since the students cannot predict the date this is a contradiction — a surprising exam IS possible by this definition….Since the students cannot predict the date this is a contradiction — a surprising exam IS possible by this definition.” I never said a surprise it not possible. I said a surprise cannot be guaranteed.

26. 26 26 Ken B

@TB:
I think we are saying the same thing (I’m just trying to avoid actually calculating the bits because I’m lazy). If you can beat the best guess from the uniform distribution (in this case; max entropy distribution generally) then you have acquired information. Which is the same as having less uncertainty. Which is what you define as being surprised. But that rather precise definition does not correspond to the loose defintion of the ‘paradox’. In particular the BobCarolTedAlice argument does not work with that precise definition.

27. 27 27 Marcelo

When I read this post, the question that immediately came to mind is “at what moment is ‘surprise’ measured”? If it is measured at the moment it is handed out, then of course no surprise is possible on Friday, and the professor’s claim that “students won’t know exactly which day until the exams are handed out” is simply false — they will know “exactly” at the end of Thursday that the exam will be handed out on Friday, if it hasn’t already been handed out by that point.

To evaluate the “paradox” it strikes me that one must precisely define how and when “surprise” is measured. I think that ThomasBayes captures this idea perfectly in his comments, and I, too fail to see the paradox (at least when “surprise” is measured according to ThomasBayes’ method, which seems logical to me).

28. 28 28 Bob Murphy

In 2003 I used the Paradox of the Unexpected Hanging to mock mainstream economists, as was my wont back then.

29. 29 29 RF

Thank God I’m no longer in school! :-)

30. 30 30 Jonathan Campbell

Assume, as Thomas Bayes says, the teacher just draws from a uniform distribution to determine the day of the exam. I hope we can all agree that the appropriate way to describe this situation is “there may be a surprise exam next week [80% chance], and there may not be [20% chance].” So there is clearly no paradox in this case.

1) Think that by rewording the above-quoted statement, the situation can be described as paradoxical. But it seems odd to think that a paradox can be created by rewording a statement which is itself not paradoxical.
2) Think that the specification of the probability distribution is what kills the paradox. But there is always a probability distribution that the students will assign to the teacher’s claim (no matter how uninformed), so a mere specification of it cannot generate a paradox.

31. 31 31 Ken B

@Bob Murphy: What made you stop? Change of heart? The ease of the game lost it its savour?

32. 32 32 Ken Arromdee

Try a modified problem. The professor tells the students “there will be an exam this week, but the exact day will be a surprise” as in the original problem.

However, unlike in the original problem, today is already Friday.

A student reasons (as in the original problem) that an exam on Friday cannot be a surprise exam. Therefore there cannot be an exam on Friday.

Then a minute later the professor gives an exam. The student (expecting no exam because of his previous reasoning) is of course surprised by this exam.

In other words, the extra reasoning for the extra days is irrelevant to the main reason for the paradox.

Statement X is “the exam is today”. When you say person A is surprised by the date, you mean “person A cannot deduce statement X”. Putting it together, “There is an exam today but you will be surprised by it, even after listening to this statement” is “X is true, but person A cannot deduce its truth from this statement”. That statement is self-referential in such a way that X really is true, yet A really cannot deduce it. (And he can’t deduce that he can’t deduce it, etc.)

33. 33 33 Mike H

If I rephrase my argument in terms of probability, the students work out that there’s a logical flaw in the professor’s statement. Their faith in their previously infallible professor is shaken. Therefore, the probabilities of the exam being on Monday is 0.1667, Tuesday 0.1667, etc, with a 0.1667 that the exam will be the week after, or that there will be no exam. Hence, even if the exam is on Friday, there’s an element of surprise.

Similarly, if you rephrase it in terms of entropy, there are actually 2.58496 bits of entropy to be cleared, not 2.32193. On Monday, 0.263 bits disappear. On Tuesday, 0.32193, etc. On Friday, the students are on the edge of their seats – will the exam be today, proving our professor wrong about the surprise? Or will there be no exam at all, proving him wrong about the exam? He’s never been wrong before. The suspense is killing us! Now it’s a 50-50 chance – a whole bit of entropy to be cleared in the next 5 minutes, whatever he does!

Interesting conclusion : if the professor is trying to maximise surprise, he should hold the surprise exam on Friday.

34. 34 34 Mike H

It’s also interesting that the professor can only be right if the students believe he could be wrong.

35. 35 35 Mike H

Let’s throw a wrench in the clockwork.

A clever student spreads a rumour that he knows when the exam is. He says he went to the professor’s office, and saw the exam date circled in big red ink on the calendar. A number of students approach him to privately ask when the exam will be. This clever student swears them to secrecy, and when he’s convinced of their trust and silence, he tells them one by one.

However, he tells some Monday, some Tuesday, some Wednesday, etc. Each day, there are some students fully expecting the exam to be that day. Whatever day the exam falls, some students are not surprised.

Does this still count as a surprise exam? Or do those students who expected the exam that day have a right to appeal to the Dean?

36. 36 36 Steve Landsburg

Ken Arromdee: I think you are exactly right — the backward induction is a red herring. The problem is that the professor says “I’m going to do X but you’re not going to know it”, the students reason that he can’t do X (because they already know it) and therefore are surprised when he goes ahead and does X.

In other words, the backward induction is pure misdirection — it’s there to hide the real essence of what’s going on.

At least that seems exactly right to me at this moment. Does anyone want to argue otherwise?

37. 37 37 David

I also like Ken’s response. I remember a version of this problem being a surprise birthday party.

One thing to also consider — the announcment was given at a specific moment. As time passes and new information is available, does this negate the ability of the students to know when the exam will be given at the time of the announcement? Of course not.

As a practical note for students today — The planning at the time of the announcement simply means the exam may be given earlier rather than later and the students would do well to prevent the surprise exam from being presented before proper study. preparations have been taken.

38. 38 38 ThomasBayes

Ken and Steve: A clever student should realize that: i) the professor wants to surprise them with a test; and ii) the professor can only surprise them with a test if they aren’t expecting a test.

Won’t that cause them to expect a test?

39. 39 39 Jeffrey

Let me add another vote for “we have reached consensus.”

Here’s my summary, which is designed only to defense the claim that we actually agree.

Mike H’s first post is entirely correct: the professor contradicted himself. All that remains is the very difficult step of seeing that nothing remains to be done.

Ken did a great job simplifying the nature of the contradiction. “X is true. And yet you will not be able to deduce X.”

Finally, to rephrase the argument in what I find to be the most useful format. (I see no reason to suppose consensus will be reached on my judgement call of “most useful.”) It can’t be Friday, it can’t be Thursday, it can’t be Wednesday, it can’t be Tuesday, therefore the test is on Monday, therefore we won’t be surprised. Thus, counting implications, the professor has claimed that they will be surprised and that they won’t be surprised. With this setup, there’s no longer anything paradoxical about the fact that the students are actually surprised.

40. 40 40 csrster

“With this setup, there’s no longer anything paradoxical about the fact that the students are actually surprised.”

41. 41 41 Harold

I agree that the backwards induction is a diversion. However, I think Thomas Bayes is probably correct- if we define “surprise” in the manner he describes. It is a way of getting round the paradox by interpreting imprecise language in a particular way.

There are two statements – say 1) “I am going to punch you, and 2) you will be surprised by the punch”. The subject reasons that since they have been told of the punch, they cannot be surprised by it. They then conclude that there will be no punch, and hence they are surprised when they are punched.

The reasoning seems to be: 1 and 2 cannot both be true, therefore I reject 1 (and hence 2). This makes both 1 and 2 false. This rejection then makes both 1 and 2 possible. It is only through 1 and 2 both being false that 1 and 2 can become true. This is the paradox.

There would be alternative way to process the information. 1 and 2 cannot both be true, therefore I reject 2. The subjects assume that the punch is coming, but not the surprise. No paradox. I think it is this option that applies in Mike H’s first post.

It seems to me that this does not remove the paradox – it still remains if we reject 1 instead of 2. Having an arbitrary choice does slightly reduce the elegance. Can someone explain where I am wrong, is rejection of 1 a more logical choice, or have I got it all completetly wrong?

42. 42 42 Jonathan Campbell

Harold-

“It seems to me that this does not remove the paradox – it still remains if we reject 1 instead of 2. Having an arbitrary choice does slightly reduce the elegance. Can someone explain where I am wrong, is rejection of 1 a more logical choice, or have I got it all completetly wrong?”

I disagree and think that the reasoning you’ve outlined (similar to mine above, but maybe clearer) explains exactly why there is no paradox.

Statements 1 and 2 cannot both be true, for the reasons you describe. But the potential punchee cannot just whimsically decide which of 1 and 2 to reject, he has to reject that which would allow logical consistency in his remaining views. For the reasons you describe, the potential punchee cannot simply reject 1 (and then 2) since that would lead to the view “i will not be punched” which may in fact be wrong. So the only logical choice is to reject 2 (and remain agnostic about 1, since his counterpart may or may not be lying).

It is as if I say “I am thinking of two numbers, X and Y. Statement 1 is that X = 0, and Statement 2 is that (Y/X = 5, and X=0).” (As in your case, Statement 2 encompasses Statement 1). As in your case, one must reject Statement 2, not both statements.

43. 43 43 Harold

Joathan Campbell: “Statements 1 and 2 cannot both be true, for the reasons you describe”
Ah – but they can be, if the subject responds by rejecting 1. Therefore the paradox remains possible, if not essential.

44. 44 44 Jonathan Campbell

Harold – They can only be true if they are unknown to the listener. So I should have said “Statements cannot both be both known and true, for the reasons you describe.”

The current scenario (stripped of backwards induction) is no more a paradox than the statements

“1) I am from Illinois. 2) You will be surprised when I produce a birth certificate showing that I am from Illinois.”

Clearly the resolution to these statements is to reject 2, but remain agnostic about 1. Totally analogous to the scenario under discussion.

45. 45 45 Harold

But look what happened in the story – there was a test and the students were surprised – so 1 and 2 were correct. It seems much more sensible for the students to have reacted as you suggest, and rejected the surprise bit, but it is not impossible that they wouldn’t. I still say there is a paradox, because you can construct the story plausibly, but it is not so satisisfactory.

Is it possible to link 1 and 2 so they must both be accepted or rejected? -i.e. like a boolean operator: 1) I am going to punch you AND you will be surprised by it. We can just say that you must do so, but that remains slightly unsatisfying also. I think it works if we do it this way. However, I am not really sure.

46. 46 46 Jeffrey

Two more observations: the first runs the risk of stating the obvious, while I think that the second is covering new ground.

1. There are two seemingly rock solid but contradictory arguments to be made. The first argument is Mike H’s observation that the students reasoning shows that the professor contradicted himself. The second argument is that both of the professors’ claims ended up being true, and surely true statements cannot be contradictory.

It’s very easy to convince yourself of one of these two arguments. The paradox isn’t resolved until the false argument is labeled and the error is found.

2. The error is in the second argument. The students were not surprised.

Surely, any clear definition of “X is surprising” must include “X could not have been logically deduced.” But it could have been deduced. The students first reason to a contradiction, as in the first argument. From this contradiction, any conclusion can be reached including “circles are square” and “the test will be on Tuesday.”

47. 47 47 Justin Dennahower

The exam can clearly be on friday, the teacher said it would be a surprise exam, not that when he hands out the exam THAT will be the surprise. So if you get to thursday afternoon then SURPRISE the exam is on friday. Its linguistic nonsense. The surprise occurs when the day of the exam is determined. It will be either when the exam is handed out of when you reach the end of the week.

48. 48 48 Pedro Fortuny

Hi!

The fact is that the students assume the teacher’s statement (which, as a matter of fact is double: there will be an exam and it will come as a surprise) is true. Which may not be the case: there may not be an exam (what if the teacher dies on Friday at 4AM or the country goes into war?) and even if there is, it may not be a surprise (but read on for this).

However, one may also think that the actutalization of future as PRESENT is always a surprise (even though we take it for granted). So, if there is an exam it is a surprise.

49. 49 49 John Haugeland

It seems to me that there’s the relatively basic failure of assuming that because something may be known thursday with knowledge accumulated explicitly through time’s passage, that that knowledge may be affirmatively asserted before the time has passed which validated said knowledge.

This seems to me to be confusing guesses about what will happen with induction. Correctly stated, the form should read “If by Thursday no exam has yet occurred, then Friday may be asserted as the only remaining option.”

The express inclusion of temporality excludes this error as possible. This is, in the end, a simple issue of clarity.

50. 50 50 nope

“There WILL be a SURPRISE exam…” is a self-referential contradiction in the spirit of “this sentence is false” (thanks prof., it’s not a surprise now). The only thing that can be concluded is that mathematicians have a corner on dead-pan literalism. Only a mathematician would ACTUALLY get stuck in an infinite loop.

51. 51 51 marris

Isn’t the lesson here that we must be careful mixing backward induction and conditional knowledge? It’s OK to induct back from something that we *know* is the end state, but not from a state which is *conditional* on other criteria unless in doing so, we keep the same conditions.

The first step of this proof shows that the exam cannot be given as a surprise on Friday, conditional to it not having been given MTWR. When we move to the next step, we cannot take this result with us. The next step assumes a conditional that the exam was not given MTW. These conditions are NOT THE SAME! We cannot rely on what we proved for Friday! Since we can no longer rely on it, we get back our two possible outcomes (the exam may be given on Thursday or Friday).

Of course, by the time we finish with Monday (and get back to the time that the teacher is talking to the students), all five days are open! So a statement “you students currently don’t know what day the quiz will be” is accurate. A statement “on the morning of the quiz day, you students will not know whether the quiz will be given” will be true for all days except Friday, conditional on it not having been given MTWR. Which is exactly what we proved with our “base case” above!

52. 52 52 Dmitriy Likhten

Ok here’s the simple answer I think works quite well: The exam is a surprise because the students don’t a-priori know what day the exam will be. Therefore they must study on the weekend. If the exam is on Friday, then it is not very surprising, however they had to study on the weekend anyways since they did not know if it was Mon – Thurs. The surprise part is not the student not knowing which day it is on the day before, it is not knowing which day it is on the prior Sunday.

Basically the exam is: Roll a 5 sided die, that is the day of the exam (1-mon, 5-friday) the randomness is the surprise.

53. 53 53 Joe User

Mistake number one is to assume that natural language uses a bivalent model, or that a word has a precise definition that cannot possibly change with context. This is clearly not the case given our ability to argue about what the value of is, is.

Mistake number two is to assume that just because the quiz is a surprise today, doesn’t mean that it will still be a surprise come friday morning when it hasn’t happened yet and must(?) happen.

Is it still a “surprise” if we can expect it the night before? Who cares? It stopped being a “surprise” the minute it was announced that it was coming. At that point, it comes down to how much of a “surprise” it will continue to be as time passes.

54. 54 54 Bob

If you give an exam on a day that it’s “impossible” to give an exam, then it’s a surprise. Since it’s impossible to give a surprise exam on Friday, and you give it anyway, then it’s a surprise.

Since you can rule out any day through induction, yet the exam can be given on any day, then you really cannot rule out any day by induction.

You certainly cannot rule out Tuesday (since the students were surprised by the exam given on Tuesday). By forward induction, you therefore cannot rule out Wednesday, Thursday, or Friday.

55. 55 55 grover

> In a class that meets every weekday morning, the professor announces that there will be an exam one day the following week, but that students won’t know exactly which day until the exams are handed out.

Reminds me of quantum physics. The two quantumly linked properties are the truthfulness of the professor’s statement and the time remaining until the last possible time for the test. When the professor makes the statement at least 5 days before the Friday, it’s true. By the Friday, the statement has become false. Around the Tuesday, the statement is half-true and half-false, an uncertain state.

> …to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached.

I’d think far more than a hundred papers have been published on the paradoxes of quantum physics. Can I suggest the surprise test paradox and quantum physics have exactly the same underlying principles behind them.

56. 56 56 Pierre B.

I think I have a way to clarify further the situation. Here it is.

I summon you to my lab. Screwed on a table, so that you can’t move them, are two wooden box with a hinged lid, labeled 1 and 2.

“Here are two boxes, one of which contains a marble.”, I tell you. “My assistant put the marble in, I have no idea in which. But I can tell you that there is a marble in one box. I have good instruments and failsafe appartus verifying it. It is an absolute certainty. You will open the box in order in a shoirt moment, but listen to me: I have studied the situation for the longest of time and have come the the following irrefutable conclusion: before you open the first box, you won’t know in which box the marble is.” You nod in agreement. This seems obvious to you so far. You have no idea where the marble is.

I continue, “Furthermore, my deep probing of the situation and commandeering of advance logic has allowed me to conclude that when you open the second box (i.e. after opening the first box) you will know if that second box contains a marble or not, since you will have seen the content of the first.” You reflect on this for a brief moment and nod in agreement. Obviously, you are much more brilliant than me and come to this conclusion rapidly.

(Now, I think everyone would agree that both of my conclusion are irrefutably correct. Here’s the wrench…)

My assistant passes by. He look at you in the eyes and proclaim “I have placed a single marble in one of those box as stated and irrefutably controlled byu the instruments. Yet, I have placed the marble one of the boxes so that you will not be able to know if there is marble in the box for *both* boxes.” He then leaves.

You are stuttering a bewildered. “Is he crazy? What he just said is impossible. When I open the *second* box, surely I will know if there is a box in it.”

Now what? How can the assistant statement be true? How is his statements different from the teacher’s? Yet, why would his statement cause you to not know the content of the second box when you open it? Why would his statement make you know the content of the first box in advance?

My take on this is as follow (taking assistent and teacher to be equiva;lent):

1. The assistant statement, taken as a whole is false.

2. The assistant statement, taken for the last box, is clearly false (and is the cause for the overall statement to be false).

3. The assistent statement, taken for teh first box is true. You won’t know if there is a marble or not in the first box.

4. The cause of the paradox for the student’s is due to causality. The inductive reasoning is going backward in time, yet causality goes forward in time. Thus Bob’s conslusion is right (the exam can’t be on Friday) but all others are wrong.

5. Thus I refute the conclusion that the one-day-only version of the paradox works. If there is a single day, there cannot be surprise on any of the days, thus the exam cannot be a surprise. When the exam is given, it won’t be a surprise.

(And I reject any claim that on a one-day version, students would be surprised by an exam that can only happen on that day. Saying “… and the student were surprised” doesn’t magically make it so.)

57. 57 57 morris

An exam on Friday is a surprise – the students don’t know whether there is an exam or not.