

The "Surprise Exam" paradox - ColinWright
http://www.thebigquestions.com/2011/12/12/the-surprise-exam-and-more-surprises/

======
brudgers
Like many philosophical puzzles, this one is based on mapping [torturing]
ordinary language into a formally logical model.

In this case "surprise" seems suspect since it is an emotional and physical
response - things which are likely to be difficult to predict using formal
systems.

Then of course, there is the fact that the surprise is announced before hand
rather than exclaimed - "Surprise!" - at the actual event as is the case with
a surprise party...and of course surprise parties often do not surprise the
person for which they are planned - indeed, sometimes forewarning is a
necessary part of the surprise party plan.

Just as a surprise party is rated a success based upon the general standards
for parties even when the person for whom it is thrown is not actually
surprised (though acting surprised is often helpful), a surprise exam is
successful based upon the relevant academic criteria. One simply does not fail
a surprise exam because they were expecting it.

What the professor announced was essentially, "On one day next week, there
will be an exam. I am not telling you which day that will be." Whatever
paradox we imagine finding is dependent on us deeming the purpose of the exam
to be to cause surprise among the students, rather than getting them to study
or measuring their knowledge or any of the legitimate purposes of the context.

Of course this does not make as interesting a story.

But it is hardly news that a flawed algorithm may produce flawed results.

The problem is that we are trying to treat ordinary language as if it was
psuedocode.

    
    
            For i = 5 to 1
            i = i - 1
            if i = 1
            then "No Exam"
            else next i

~~~
Dylan16807
The way to put the problem of surprise into logic is more like "On one day
next week there will be an exam. It will not be possible to logically
determine which day the exam is on." It's a very interesting story, you just
have to preserve the core of the problem.

~~~
brudgers
> _"It will not be possible to logically determine which day the exam is on"_

Once that becomes a premise, the efforts of the students are of the sort who
should be logically classed with:

 _persons in a state of insanity, whose brains are so disordered and clouded
by dark bilious vapors as to cause them pertinaciously to assert that they are
monarchs when they are in the greatest poverty; or clothed [in gold] and
purple when destitute of any covering; or that their head is made of clay,
their body of glass, or that they are gourds..._

[Descartes]

~~~
Dylan16807
Well at the start of the argument they're not trying to determine the day it
is on, but rather trying to find one day that it is not on.

------
raganwald
I first read of the ‘paradox’ in one of Raymond Smullyan’s excellent books. My
favourite formulation goes like this:

A philosophy professor tells the class there will be a surprise exam this
week. He and the class discuss the exam, and they ‘prove’ that there can be no
surprise exam. When everyone has accepted the proof, he announces that there
will be an exam on the spot! The exam has only one question: Prove that the
exam is indeed a surprise, invalidating the proof that there can be no
surprise exam.

This formulation suggests that the exam is a surprise if the students convince
themselves that there can be no surprise exam, which is an entertaining trail
to follow.

Smullyan also tells an anecdote from his childhood. He and his brother liked
to ‘fool’ each other on April Fool’s, and one year his brother announced that
the coming April Fool’s Day, he would fool Raymond as he had never been fooled
before. Well, the day came and Raymond steeled himself for the prank. He
waited, expecting a prank or lie with everything they did and every
cinversation they had, but no prank, no lies, no fooling.

As he lay in bed that night, Raymond realized that his brother had, in fact,
fooled him. Or had he? How could not fooling someone constitute fooling him?
According to Raymond, puzzling over the nature of ‘fooling’ was part of his
lifelong journey into philosophy and logic.

~~~
frooxie
"When I came home I expected a surprise and there was no surprise for me, so,
of course, I was surprised". — Ludwig Wittgenstein

~~~
raganwald
Dictionary author Noah Webster was in his office with his secretary, engaged
in activities well outside of her professional duties. Noah’s wife opened the
door and exclaimed in shock: “Noah! I’m surprised!!”

Noah looked up. “No, my dear, you are amazed. It is we who are surprised."

------
rcthompson
There is a guessing strategy that the students can use to guarantee that the
exam will not be a surprise. Each day, before class begins, the students
declare "We conclude that the exam will be today with probability 1.
Therefore, we have advance knowledge of the date of the exam." They might be
wrong a few times, but they will be right on the day of the exam, so the exam
will not be a surprise.

This means that the professor cannot guarantee a surprise exam, so his
original claim is false. But note that the students' strategy requires them to
always assume that the exam will be on the earliest possible date and to plan
their study time accordingly, which is probably the behavior that the
professor was trying to encourage anyway. It just so happens that he has to
lie about the surprise exam in order to do so.

~~~
shasta
Belief probability isn't an arbitrary number that you get to declare. A
rational student isn't going to be 100% certain that the test is Monday. If
you think you could be, let's set this up. Bring your wallet so that we can
wager.

~~~
rcthompson
Yes indeed, my suggested strategy assumes that there is no penalty for
guessing wrong except on the day of the exam.

------
jonshea
Before posting that there is an easy or obvious resolution to the “paradox”,
please read at least the first couple pages of the literature survey that
Landsburg links to [1].

""" The meta-paradox consists of two seemingly incompatible facts. The first
is that the surprise exam paradox seems easy to resolve. Those seeing it for
the first time typically have the instinctive reaction that the flaw in the
students’ reasoning is obvious. Furthermore, most readers who have tried to
think it through have had little difficulty resolving it to their own
satisfaction.

The second (astonishing) fact is that to date nearly a hundred papers on the
paradox have been published, and still no consensus on its correct resolution
has been reached. The paradox has even been called a “significant problem” for
philosophy [30, chapter 7, section VII]. How can this be? Can such a
ridiculous argument really be a major unsolved mystery? If not, why does paper
after paper begin by brusquely dismissing all previous work and claiming that
it alone presents the long-awaited simple solution that lays the paradox to
rest once and for all? """

fn 1: <http://www-math.mit.edu/~tchow/unexpected.pdf>

------
DrJ
couldn't they just say, assuming iid:

    
    
      on Thursday you know that the probability that the exam will be on Friday is 1
      on Wednesday you know that the probability that the exam will occur Thursday is .5
      etc.   etc.
      on Sunday you know that the probability that the exam will occur on Monday is .2
    

So if you look at it from (naive) probability, you can never make the 2nd
assumption that on Thursday you will not have surprise test.

you could say the pmf is: .2, .5 .8 .17 .5 and the cdf is .2 .25 .33 .5 1
Which from memory looks like a Beta Distribution.

~~~
prof_hobart
No they couldn't. If the statement was "the exam is going to be on a randomly
selected day next week", then you'd be correct.

But that's not the case.

The teacher claims that "students won’t know exactly which day until the exams
are handed out", whereas by the end of Thursday, they would be 100% certain
that the exam was going to be on Friday. Therefore it is impossible for a
surprise exam to be on the Friday, so that can be ruled out as a possibility.

Which means that by the end of Wednesday, the only possible day for a surprise
exam becomes Thursday, and the same logic which ruled out Friday then applies
to Thursday (if it has to be Thursday, it can't be a surprise, so it can't be
Thursday), and all the way back through the rest of the week.

~~~
pierrebai
There a few traps in the question, leading to different refutation or
explanation.

The first is the ambiguous definition of surprise. This lead to easy
explanation of the paradox. The teacher might as well hand out an exam on a
red piece or paper and claim that _that_ is a surprise.

The second is that the teacher invalidly presumes to be able to predict the
mental state of his students. This can be stated simply: over the weekend, all
his students die unexpectedly. The following week, none of his student are
surprised since the set of surprised students is empty.

We can work around those two flaw by ignoring the no-student case and assuming
that "surprised" is replaced by a more exact formulation: "on the day the exam
occurs, nobody would be able to say ``I knew the exam was today''".

In this formulation, I contend that nothing can be concluded.

The reason for this is that induction can only be applied on facts. The future
is not a fact, so to apply induction on a sequence of future events is an
error. The only conclusion that one can reach is the contrafactual: if we were
Friday then the exam cannot occur on Friday. This is because the reasoning
does not rely on any inductive argument but rely only on things that would be
facts inside the contrafactual: if we were on Friday now, it would be false
that the exam cannot be held on that day and that nobody could say that the
exam is to be held that day.

A contrafactual of this nature cannot be built for Thursday, since in _that_
contrafactual ("if we were Thursday...") one would have to reason by induction
about teh future. Of course, in the Thursday-contrafactual, one could state
the Friday-contrafactual, but you can't build inudction on contrafactual (i.e.
non-fact).

~~~
prof_hobart
>The first is the ambiguous definition of surprise.

He doesn't actually use the word "surprise" anywhere in his statement, and
spells out pretty clearly what the nature of the surprise would be - "students
won’t know exactly which day until the exams are handed out. Nothing, other
than coming in on a particular day, not knowing for certain that the exam will
be that day, but getting the exam on that day nevertheless, would do - e.g.an
exam on the Friday would not qualify, as you would already know that this was
the day of the exam. And nor would red paper.

>The following week, none of his student are surprised since the set of
surprised students is empty.

Again, he doesn't claim that anyone would be surprised. He said that they
"won’t know exactly which day until the exams are handed out." I suppose you
could argue that if they are dead, then they won't even know on that day
either, but that's a pretty tenuous point and anyway technically he doesn't
explicitly claim that they would know at that point, just that they wouldn't
beforehand.

>"on the day the exam occurs, nobody would be able to say ``I knew the exam
was today''".

Which is pretty much what he did say.

>A contrafactual of this nature cannot be built for Thursday

Why not? The argument against it being on Friday (and I think we're both
agreed that it's impossible for it to be on Friday and still line up with his
statement) is that if there's only one possible choice for what day it's on,
then the pupils - or at least any that are still alive - would know that the
exam was on that day. And if it can't be on Friday, then when it gets to first
thing Thursday how many choices are left for possible days for the exam to
line up with his statement? One - so we're back to it being a certainty as to
which day it would be.

~~~
pierrebai
True, the way it was worded not involve surprises. Instead, the ambiguity is
shifted to the meaning of what consitute knowing, which is pretty much
equivalent and causes the same confusion. My aim was to try to put aside such
semantic argumentation.

My main point was that future events can't be held as facts.

I used the death of all students as an example. I'm not sure why this would be
a tenuous point: it's my central thesis.

Re-reading the paradox as worded, I conclude that the teacher is right: the
students won't know exactly which day because they cannot know the future as a
fact. They might die, the teacher might die, the sun may explode. I don't
consider these as tenuous arguments. They are concrete counter-example that
any reasoning involving future events is flawed.

~~~
prof_hobart
It's tenuous because it's almost entirely tangental to the point of the
paradox. You could caveat his statement with "assuming that we are all still
alive, and in a position to take an exam on any given day next week..." or
something like that, and it doesn't change the actual point of the paradox one
bit.

Trying to explain away this paradox by talking about people possibly being
dead by next week is about as useful as taking "This statement is false" and
explaining it away by saying that it could be in a language where "false"
means true.

~~~
pierrebai
Then stop focusing on "being alive" bit. I was giving this solely as an
example of why making statements about the future and basing arguments about
some future state cannot work. Yet you take it as some trick to avoid having
to explain the paradox.

(Your analogy with pretending that the word false doesn't mean false is
entirely off base. I don't see the correspondence. But no matter.)

I think I've found a formulation that won't offend your sense: due to time's
arrow and causality (which both go forward in time), one cannot make an
induction that goes backward in time. This would cause a circular dependency
in the inductive proof.

In this case, the steps of the inductive proof go from Friday, down to
Thursday, Wednesday, etc. Yet at the same time, causality goes from Wednesday
to Thursday to Friday. One cannot simply dismiss causality.

IOW, we have the inductive process N, N+1, N+2, where N is used to proove N+1,
etc and where N = Friday, N+1 Thursday, all the while N state depends on N+1
because due to real-world causality, what happens on Friday depends on
Thursday.

You seem to wish to deny causality.

~~~
prof_hobart
I'm not the one focusing on it. I'm saying it's pretty much irrelevant to the
point of the paradox. It's not about being able to predict any future event
except the date of the exam - as I've said, you could put all of the caveats
you want in there about any other events either happening or not happening
(e.g. "assuming that we're all still alive, and assuming that they don't
rename the days of the week before next Friday, and that no-one's changed the
terms of the teacher's statement etc") and it doesn't change the paradox.

As for causality, I don't really see how that's relevant either. Assuming
you've accepted whatever caveats we feel like adding, then nothing (apart from
the test having happened) that happens on Monday, Tuesday, Wednesday or
Thursday is going to change the fact that if you've got to Friday without the
test having happened you'll have certainty that the test will be on Friday,
and therefore won't be a surprise.

In other words, the only things required for the paradox to be a paradox are
the idea of a test happening on a day that the pupils can't predict and a
deadline for when that test must happen by.

(And the reason my analogy about the word false is relevant is that it's
pretty clear what the bounds of both paradoxes are, and you "solve" either of
them by throwing in things - whether that's events external to the point of
the paradox, or word definitions external to the point - that are outside
those bounds, but in either case it doesn't tell you anything useful about the
paradox itself).

------
ghshephard
One cannot be surprised by a P(1) event. Ergo, the initial statement, that the
students will be "surprised" by an event that is guaranteed to happen next
week (the exam) is the source of the logic chain that ends in Reductio Ad
Absurdum. There is a collapsing probability curve as to _when_ the exam will
occur, with an even distribution (from the perspective of the students) of
P(.2) for any given day starting on Sunday night, and collapsing to P(1) for
Friday after Thursday morning.

Clever logical conundrum though. Certainly forces you to think deeply on words
like "Surprise."

------
vorg
> 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.

------
simon_weber
I highly recommend Poundstone's Labyrinths of Reason for an entertaining
coverage of this and other similar problems.

------
davnola
The truthmaker for whether it is a surprise or not clearly depends on their
own belief states. They are reasoning about what they would or would not
believe in different circumstances, so they should include a premise about
their belief states in their induction, otherwise their reasoning is unsound.

They reason, if there is going to be an exam, and it hasn't happened by
Thursday, then it will happen Friday, and _we would believe_ it would happen
on Friday and it cannot be a surprise.

They _should_ reason like this: if there is going to be an exam _and we
believe there will be an exam_ , and it has not happened on Thursday &c then
it cannot be a surprise.

But, they stopped believing the exam was happening. Doh.

EDIT: formatting

------
antonios
My thoughts:

The exam can be characterized as "surprising" until the end of Thursday's
class. Getting the probability at the end of each day for the exam to be made
from now on, we have (M=Monday, T=Tuesday etc):

End of Sunday : P(exam=M,T,W,T,F|S)= 1/5

End of Monday : P(exam=T,W,T,F|M)= 1/4

End of Tuesday : P(exam=W,T,F|T)= 1/3

End of Wednesday : P(exam=T,F|w)= 1/2

End of Thursday : P(exam=F|T) = 1

So, after the end of Thursday class, the probability of the exam is 1, so the
students cannot be surprised any more as they are sure that they will be
examined Friday. This means, that in every day except Friday the students can
be surprised by the exam.

Anybody to point me where the above interpretation is wrong?

------
lurker17
Setup

1: "You will be surprised to learn the truth value of this is statement, or
this statement is false."

Case A

2: "I will not be surprised. I know that the statement is true."

1: "Then the statement is false, and you have been surprised!"

Case B

2: "I will not be surprised. I know that the statement is false."

1: "Then the statement is true, and you have been surprised!"

Case C

1: "I am surprised!"

2: "The statement is true."

------
shasta
The paradox is most interesting after only one day remains. On Friday, after
Monday through Thursday have passed with no exam, the professor's declaration
becomes "we will have an exam later today, but you will not know it until
then." If the professor tells you this, do you know the exam will happen? If
you can know it, then the professor was wrong (at least) about the second half
of his prediction, which brings into question the first half.

------
Symmetry
It seems like there might be an isomorphism between "You can't know the exam
will be tomorrow" and "You can't know this sentence is true."

------
fab13n
The trick is that "being surprised", or more accurately "not knowing on which
day the exam will take place", is changing over time.

"D" being the day at which the exam takes place, "not knowing D on Monday" is
not the same proposition as "not knowing D on Thursday". If you change the
proposition tested at each inductive step, your induction is invalid.

------
grandalf
Isn't there a problem with the first student's assertion that by Friday the
exam would not longer be a surprise? The definition of a surprise in the
context of the question is in relation to the state of the students' brains at
the time the professor made the announcement.

------
gldalmaso
Perhaps an announced 'surprise' ceases to be one, but none the less, what the
teacher ought to have said is that there _will_ be a test in a _randomly
picked_ day next week. All that's left for the students to guess is how many
days they have to study.

~~~
Dylan16807
Not when the teacher is deliberately invoking paradox.

------
mikeash
Can this be resolved by rejecting the idea that every statement _must_ be one
of true or false? If the professor's statement that there will be a surprise
exam is neither true nor false, but merely not yet decided, then there is no
contradiction. Indeed, the professor can't know for sure that his exam will be
a surprise, as the students may have guessed, or placed a spy camera in his
office, etc. The professor can guarantee that the exam is not a surprise by
announcing it, but he can never guarantee that it is a surprise, only make a
good attempt.

Similarly, "this sentence is false" is neither true nor false, but just
nonsense. "The set of all sets that do not contain themselves" is not a
paradox or a contradiction, it simply describes an object that doesn't exist.
Likewise, the professor's statement isn't known to be true or false at the
time it's made, and either outcome is possible.

~~~
davnola
Perhaps it can, but the challenge then is to identify what's different about
the professor's statement from a statement like "It's raining". Why doesn't
the first have a definite truth value, but the second does? (What the
professor himself believes is not relevant to the actual truth of his
statement.)

You might be interested in verificationism
[http://plato.stanford.edu/entries/logical-
empiricism/#EmpVer...](http://plato.stanford.edu/entries/logical-
empiricism/#EmpVerAntMet) (apologies if you know about it already).

~~~
dspeyer
What's different about it is its circularity. The students' anticipations of
the exam are based on their anticipations of the exam. Sort of like "this
statement is false".

~~~
davnola
The students' anticipations of the exam _are_ their anticipations of the exam.
I don't see the circularity yet. It's clearer in the Liar Paradox.

~~~
ajuc
Circularity is here:

    
    
        P( exam is today | P(exam is today | exam was not earlier) = 1) = 0
    

EDIT: if this is good formalisation of proffesor sentence. I'm not 100% sure.

------
danso
I liked this blog post enough that I clicked through to the author's book, but
it's gotten pretty middling reviews on Amazon: [http://www.amazon.com/The-Big-
Questions-ebook/dp/B002T0I02Y/...](http://www.amazon.com/The-Big-Questions-
ebook/dp/B002T0I02Y/ref=kinw_dp_ke?ie=UTF8&m=AG56TWVU5XWC2)

------
its_so_on
This "paradox" is no paradox at all, and has a ridiculously easy and obvious
resolution.

As a teacher:

1) you CAN guarantee for your students that there will be a pop quiz Monday
through Friday of next week.

2) You CANNOT _ALSO_ guarantee for them that they will be surprised that day.
(i.e. that they won't be sure, on that day, that it's that day).

This is because if it hasn't happened by the last day it can happen, it will
happen the last day it can happen.

This obviously makes sense. There's no paradox. There's nothing difficult.

You simply cannot GUARANTEE them surprise, since out of whatever possible
space of days it can happen, on the last day it won't be a surprise.

Whether that space is 1, 2, 3, 4, or 5 days, on the last of that day it is not
a surprise.

For any space of days in which the exam can happen, the last day would not be
a surprise;

therefore, you cannot guarantee them both that there will be a space of days
in which it can happen, and also GUARANTEE them that it will be a surprise.
(in every eventuality. You can make them an 80% guarantee that it's a
surprise, while making them a 100% guarantee that it's a quiz between monday
and friday of next week. if you can increase the number of days you can
increase your guarantee further and further. if I guarantee you a quiz on one
of the next 100 days, I can also 99% guarantee you that you won't be sure
you're having it that day -- i.e. will be 'surprised' -- on the day that I
give it.)

prior probability. pick a day monday to friday. quiz time comes. monday to
thursday they're surprised; if it's on friday, they're not surprised.

why you would think you could also surprise them after giving them a chance to
"open" every day up to then is beyond me.

THis is like giving you a guarantee: here are five envelopes, one has a heart
in it. I guarantee you will be surprised when you open the one with a heart in
it, even if you've opened four of the five envelopes already and found them
empty.

um....no... you can't guarantee that. it makes perfect sense, and there's no
paradox or worth wasting any breath over.

~~~
tolmasky
Your envelope example is great because it sheds even further light on the
problem: the surprise does in fact exist, it just exists before the moment the
final event takes place. In your envelope example, once you open the fourth
envelope and discover nothing inside it you can _simultaneously_ be surprised
that it wasn't there and that the fifth envelope _will_ contain the heart. In
other words, the two events collapse into one. Similarly, when going to class
on Thursday you will be "surprised" to find that the test is not that day but
rather the next. The only thing that makes the exam version "confusing" is
that you then have a full 24 hours to ruminate over this fact, thus making it
_seem_ like its not a surprise at all -- but its actually equivalent to on the
first day being told when the test will be that week (at which point no one
would rule out Friday) - it may be surprising at that moment but clearly won't
be by the time the test actually takes place.

~~~
its_so_on
right, exactly. This whole problem collapses if you don't 'open the envelopes
one at a time' while having my 100% GUARANTEE that you won't be sure there's a
heart in the one that does have one, after opening any others you've gone
through.

An 80% guarantee is fine, but 100% guarantee that you won't be sure, is
incompatible with having a space of envelopes and going through it one by one.

why would anyone think otherwise?

the way in which I discovered this resolution is by coding up a perl script to
'monte carlo' different scenarios. I realized at the location that I made the
teacher actually have to choose which day the exam will be, there is a space
of days.

(they have to choose - or end up choosing - as they break their first
guarantee if they don't have it on any day of the week or more than one day of
the week, or whatever. it has to be one and only one day of the week, however
they end up getting there.)

it doesn't matter if the teacher is choosing one of five days, 10 days, 100
days, whatever. The methodology is that the students get to go through the
days one by one.

If you are choosing one of five days, there is a 20% probability of choosing
the last of them; therefore given this methodology you can only 80% guarantee
them surprise.

if you are choosing one of ten days, there is a 10% probability of choosing
the last of them; therefore given the methodology you can only 90% guarantee
them surprise

if choosing one of 100 days, 1% probability you chose the last of them; you
can 99% guarantee them surprise.

Bottom line, which you discover if you code it up to run in simulations: at
some point the teacher MUST actually choose a day to meet their first
guarantee. (If they choose none - or end up giving it on none - then they've
broken their first guarantee. Any algorithm that doesn't end up choosing a day
100% of the time is wrong).

At the point of choosing a day, it doesn't matter if you make the Teacher
choose monday through thursday, monday through wednesday, monday or tuesday,
or hard-code it to choose Monday. Whatever the teacher chooses, if the
students have access to the teacher's algorithm (which describes the space of
possible days), then the last day of the space they would not have surprise.
The teacher can thus give a guarantee equal to the number of days OTHER than
the last day, over the number of days in the teacher's space. In the usual
sense, this is 80% guarantee of surprise.

of course, another question is if the teacher gets to follow an algorithm the
students must only guess at. (They don't know what his algorithm is).

In this case, for your simulation you can just hard-code the teacher always
giving the test on Monday. Since the students don't know this is his
algorithm, they will be "surprised" (they could have thought that he was
picking from one day monday thru friday, they had no way to be sure that he
was hard-coded to pick Monday), thus fulfilling both criteria of not being
sure of what day the test is, and being given a test monday to friday.

sonofabitch. new solution: The teacher can meet his obligations by being hard-
coded to give the test on Monday, but not telling his students that this is
the algorithm. Since they could be thinking he's choosing from a space of
monday thru friday, they could think that the last day is friday: it's really
Monday.

this is an interesting aspect I hadn't considered (a secret algorithm).

In this case I would say my response is more nuanced: 1) To whatever extent
the students get to know of the teacher's algorithm for picking days, they are
that much less able to be given a guarantee of 100% surprise on the day of the
exam.

Therefore, if the teacher is completely secret about his algorithm (i.e. he is
NOT 'picking randomly monday thru friday') he can surprise the student. Any
information he gives his students about his algorithm takes away from the
extent to which he can guarantee their surprise. EVEN IF the algorithm
includes randomness.

In other words: if the students know the teacher is picking a random day
monday thru friday, then he cannot offer a guarantee of surprising them.

if the students have no idea what algorithm the teacher uses to pick a day
(and, therefore, have no way to guess at the space), he can offer a guarantee
of surprising them.

Thus if students know nothing at all about the teacher, and he knows this
fact, he can meet his obligation by giving the test from a space of one day
(Monday) without any randomness or space of possibilities.

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georgieporgie
This is why I hated philosophy in college. The logic is extremely simple. The
difficulty comes in ridiculously pedantic interpretations of the English
language.

Give me mathematics any day. Philosophy? You and the lawyers can keep it.

