
Which answer in this list is the correct answer to this question? (2017) - akakievich
https://math.stackexchange.com/questions/2217248/which-answer-in-this-list-is-the-correct-answer-to-this-question
======
lovasoa
Here is a single python statement that solves the problem:

    
    
        print([
            q for q in itertools.product((True, False), repeat=6)
            if q == (
                all(q[1:]),  # 1. All of the below.
                not any(q[2:]),  # 2. None of the below.
                all(q[:2]),  # 3. All of the above.
                any(q[:3]),  # 4. One of the above.
                not any(q[:4]),  # 5. None of the above.
                not any(q[:5]),  # 6. None of the above.
            )
        ])
    
    

[https://gist.github.com/lovasoa/f2b4ed93e755bf4172583d28f206...](https://gist.github.com/lovasoa/f2b4ed93e755bf4172583d28f20635a3)

~~~
Recursing
A less clever solution, using the wonderful z3

    
    
      import z3
      answers = [z3.Bool(f"answer{i}") for i in range(1,7)]
      implications = [
          z3.And(answers[1:]),         # All of the below
          z3.Not(z3.Or(answers[2:])),  # None of the below
          z3.And(answers[:2]),         # All of the above
          z3.Or(answers[:3]),          # Any of the above
          z3.Not(z3.Or(answers[:4])),  # None of the above
          z3.Not(z3.Or(answers[:5]))]  # None of the above
      constraints = [z3.Implies(ans, impl) for ans,impl in zip(answers, implications)]
      z3.solve(constraints)

~~~
colanderman
Not enough constraints. You must (EDIT: should? see discussion below) also
constrain that there is exactly one `answer{i}` which is true, and all the
implications should be equalities (else, for example, 6 is a valid answer,
even though that would imply 5 is true and thus contradict 6).

It just so happens that the first result Z3 finds is the correct one. But if
you exclude that result with an additional constraint, it will find another.
(This is general is a good way to check your work.)

(I just did the same exact exercise with Z3 and CVC4, but using SMTLIBv2
syntax.)

~~~
arcatek
I don't think there is any constraint that a single answer is true, otherwise
the exercise wouldn't list "All of them" as a possibility.

~~~
colanderman
I feel like it's implied by the question itself? "Which answer [singular] is
the [definite article] correct answer"

But you may be right. I detest word puzzles for exactly this type of
ambiguity.

Same goes for implication vs. equality. Why "should" 5 being true contradict 6
being correct? Just because 5 happens to be true doesn't necessarily mean it
is "the" "correct" answer. The "real" answer depends on an interpretation of
the English-language formulation that _most_ people will apply, but not all.

~~~
mafuy
The puzzle itself contains enough contradictions that a single answer is
forced. So it is unclear from the question, but the result is definite: There
is exactly one answer. (Of course, that fact is not obvious.)

------
dtjohnnyb
Similar one I enjoyed in the Naive Bayes article from yesterday
[https://blog.floydhub.com/naive-bayes-for-machine-
learning/](https://blog.floydhub.com/naive-bayes-for-machine-learning/)

    
    
      Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?  
      A) 25%  
      B) 50%  
      C) 60%  
      D) 25%

~~~
nothis
What's the correct answer?

~~~
vesinisa
Undefined. It's a paradox.

------
jetrink
I've always thought of these as "Wayside School" problems, because I first
encountered them in the children's book, Sideways Arithmetic From Wayside
School by Louis Sachar. I highly recommend it for any child who is into
puzzles. I really enjoyed how the book deconstructed the format of tests and
quizzes that I was so familiar with at the time. My favorite section was the
sideways math, where words are added or multiplied and you had to deduce the
values of the letters. E.g.,

    
    
          D O G
        x   A D
        _______
        
          D O G
        A D O
        _______
        
        A G O G
    

Maybe children don't learn to do arithmetic like this anymore and this section
will be utterly impenetrable to them.

~~~
aj7
Since we’re off subject, and Homer Jones made the NFL’s best something, I
offer: FRAN + JONES = SCORE

------
lovasoa
This is the kind of things that are a real pleasure to model in Haskell. Here
is a nice solution:

    
    
        solve funcs = filter (matches funcs) (possibilities funcs)
        matches funcs truths = truths == zipWith (($).($truths).flip) funcs [0..]
    
        possibilities [] = [[]]
        possibilities (x:xs) = concatMap (\x -> [True:x, False:x]) (possibilities xs)
    
        all_of = ((and.).)
        one_of = ((or.).) 
        none_of = (((not.or).).)
    
        the_above = take
        the_below = drop.(+1)
    
        main = mapM_ putStrLn $
          zipWith
            (\i s -> "Answer number " ++ (show i) ++ " is " ++ (show s))
            [1..] $
            head $ solve [
              all_of  the_below,
              none_of the_below,
              all_of  the_above,
              one_of  the_above,
              none_of the_above,
              none_of the_above
            ]

~~~
lovasoa
Run it online at:
[https://repl.it/repls/WeeSlightExams](https://repl.it/repls/WeeSlightExams)

------
jonathanstrange
It seems that 5 is a winner, as stated in the accepted answer.

That's a pity, first I thought the goal of the question was to create an
instance of Yablo's Paradox. Yablo's paradox is important because you cannot
just argue that the paradox arises from an obviously incorrect definition, as
people sometimes do for classical semantic paradoxes.

~~~
cyborgx7
It would be very boring if it was paradoxical.

It would just be a more complicated version of:

1\. Statement 2 is wrong.

2\. Statement 1 is wrong.

Edit: I just looked up Yablo's paradox. It only works because there are
infinite statements. Since there is a finite number of statements here, it
could not be an instance of Yablo's paradox. Only a circular paradox.

~~~
triska
One interesting aspect of this concrete example is that even though the two
statements seem contradictory at first sight, there are in fact two
assignments of truth values to statements so that no contradiction arises.

For instance, using Scryer Prolog and its SAT solver to model the situation:

    
    
        ?- sat(S1 =:= (S2 =:= 0)),
           sat(S2 =:= (S1 =:= 0)).
    

As answer, we get a symbolic expression that compactly captures all concrete
solutions:

    
    
        clpb:sat(S1=\=S2) 
    

This means that as long as the truth value of S1 is different from that of S2,
the puzzle is solved. This is intuitively admissible, because if one of the
statements is false, then the other is true.

It would be different if for example Statement 1 said “Statement 1 is false”,
because then there is no satisfiable assignment at all:

    
    
        ?- sat(S1 =:= (S1 =:= 0)).
        false.

~~~
cyborgx7
I actually meant to have one say false and one say true. I just made a mistake
in writing it out.

------
eyko
I'm a bit confused. It seems that we're all looking for the answer that can
hold "true". That's fine, however, does 5 answer _this question_? From my
understanding, the question itself is impredicative[1].

"Which answer in this list is the _correct_ answer to _this question_?"

Being self-referential makes it difficult to decide what correct in that
context means. Does the question even allow for a no-answer/inconclusive
result?

1\.
[https://en.wikipedia.org/wiki/Impredicativity](https://en.wikipedia.org/wiki/Impredicativity)

edit: Alright, just noticed that user "DanielV" already covered its
impredicativity (assuming that's what they mean by impredictive*)

------
Gunax
For an extended version, see this:

SELF REFERENTIAL QUIZ
[http://faculty.uml.edu/jpropp/srat-Q.txt](http://faculty.uml.edu/jpropp/srat-Q.txt)

Soltion(s):
[https://faculty.uml.edu/jpropp/srat.html](https://faculty.uml.edu/jpropp/srat.html)

------
hokkos
With Z3 : [https://rise4fun.com/z3](https://rise4fun.com/z3)

    
    
        (declare-const a Bool)
        (declare-const b Bool)
        (declare-const c Bool)
        (declare-const d Bool)
        (declare-const e Bool)
        (declare-const f Bool)
        (define-fun conjecture () Bool
        (and 
            (= a (and b c d e f))
            (= b (and (not b) (not c) (not d) (not e) (not f)))
            (= c (and a b))
            (= d (or a b c))
            (= e (and (not a) (not b) (not c) (not d)))
            (= f (and (not a) (not b) (not c) (not d) (not e)))
            (xor a b c d e f)
        )
        )
        (assert conjecture)
        (check-sat)
        (get-model)
    
    

result :

    
    
        sat
        (model 
        (define-fun f () Bool
            false)
        (define-fun b () Bool
            false)
        (define-fun a () Bool
            false)
        (define-fun c () Bool
            false)
        (define-fun d () Bool
            false)
        (define-fun e () Bool
            true)
        )
    

Old code was with another tool :
[http://logictools.org/](http://logictools.org/)

    
    
      (a-> (b & c & d & e & -f)) &
      (b -> (-b & -c & -d & -e & -f)) &
      (c -> (a & b)) &
      (d -> (a | b | c)) &
      (e -> (-a & -b & -c & -d)) &
      (f -> (-a & -b & -c & -d & -e)) &
      ( a + b + c + d + e + f)
    

then

    
    
      (a <-> (b & c & d & e & f)) &
      (b <-> (-b & -c & -d & -e & -f)) &
      (c <-> (a & b)) &
      (d <-> (a | b | c)) &
      (e <-> (-a & -b & -c & -d)) &
      (f <-> (-a & -b & -c & -d & -e)) &
      ( a + b + c + d + e + f)
    

where it would crash

~~~
colanderman
Implication is not strong enough. That permits e.g. f to be true and e to be
false.

Also I don't know the specifics of the language that tool uses, but typically
variadic XOR simply constrains that an _odd_ number of its operands are true.
(It so happens that there are no solutions to this problem with 3 or 5 true
answers.)

~~~
hokkos
I changed the solver as it would crash when I corrected my code, and redone it
with z3 online.

------
phigcch
In the same vein, there's the classic infamous self-referential aptitude test.
[http://faculty.uml.edu/jpropp/srat-Q.txt](http://faculty.uml.edu/jpropp/srat-Q.txt)

~~~
piepoter
I'm a huge fan of this, if only I had some sort of class to give this test to.

------
jhoechtl
An expert system answer (Prolog, CLIPS) would be cool to see here.

~~~
tephra
Using the clpb library [0]:

    
    
      solution([A1,A2,A3,A4,A5,A6]) :-
            sat(A1 =:= A2*A3*A4*A5*A6),
            sat(A2 =:= ~(A3+A4+A5+A6)),
            sat(A3 =:= A1*A2),
            sat(A4 =:= A1+A2+A3),
            sat(A5 =:= ~(A1+A2+A3+A4)),
            sat(A6 =:= ~(A1+A2+A3+A4+A5)).
    

[0]
[https://www.metalevel.at/prolog/puzzles](https://www.metalevel.at/prolog/puzzles)

~~~
grubb
Alternatively using z3 [0]:

    
    
      (declare-const a1 Bool)
      (declare-const a2 Bool)
      (declare-const a3 Bool)
      (declare-const a4 Bool)
      (declare-const a5 Bool)
      (declare-const a6 Bool)
    
      (assert (= a1 (and a2 (and a3 (and a4 (and a5 a6))))))
      (assert (= a2 (not (or a3 (or a4 (or a5 a6))))))
      (assert (= a3 (and a1 a2)))
      (assert (= a4 (or a1 (or a2 a3))))
      (assert (= a5 (not (or a1 (or a2 (or a3 a4))))))
      (assert (= a6 (not (or a1 (or a2 (or a3 (or a4 a5)))))))
      (assert (or a1 (or a2 (or a3 (or a4 (or a5 a6))))))
    
      (check-sat)
      (get-model)
      (exit)
    

The model is satisfiable where only a5 is true, anyone can run it themselves
and play with it [1].

Additionally forcing a5 to be false by adding the following to the list of
assertions:

    
    
      (assert (not a5))
    

And the model becomes unsatisfiable.

Edit: Finally, to make sure there's not some possible solution where a5 is
true as well as another assertion you could alternatively add this assertion:

    
    
      (assert (and a5 (or a1 (or a2 (or a3 (or a4 (or a6)))))))
    

And the model also becomes unsatisfiable. So 5 only seems to be the correct
answer.

[0] [https://github.com/Z3Prover/z3](https://github.com/Z3Prover/z3)

[1] [https://rise4fun.com/Z3/1DMW](https://rise4fun.com/Z3/1DMW)

~~~
Recursing
I posted a solution using the python z3 library, which might be easier to use
for people used to python, here:
[https://news.ycombinator.com/item?id=21545143](https://news.ycombinator.com/item?id=21545143)

~~~
Recursing
That solution was not correct, see
[https://news.ycombinator.com/item?id=21568151](https://news.ycombinator.com/item?id=21568151)

------
vagab0nd
I feel the question is unnecessarily convoluted. To really answer the
question, you have to recursively refer to the question itself. So to parse
the question, I'm calling a function recursively without a stop condition.
They could have just asked, which statement below is true. And for all intents
and purposes the answer would have been the same.

~~~
zumu
I too am of the opinion the question recurses indefinitely, which would mean
it's unanswerable and perhaps not even a valid question technically speaking.

------
microcolonel
LinkedIn's skill tests are hilariously poor. They ask unanswerable questions,
and mistype the answers. The more familiar you are with the subject matter,
the more uncertain you become.

~~~
cdubzzz
I noticed those randomly one day and did a few of them. I failed the one for
the primary programming language I have worked in for the past decade and
passed ones for two different languages I have hardly used and AWS, which I
barely know anything about outside of EC2 and R53.

I felt like there was a big difference in the quality of the questions and
answers for each quiz. For the AWS one, for example, the phrasing used for
most of the questions and answers made the correct answer basically obvious
without actually knowing it to be correct.

~~~
odonnellryan
Same with UpWork. Tests on these sites are largely garbage.

------
TheGrumpyBrit
We can easily narrow down the correct answer to "none of the above", which
leaves 5 or 6. But since 5 is above 6 and is a correct answer, 6 cannot be
correct. So it must be 5.

------
davvolun
I could be wrong, but the question is ill-stated; should be something like
'which of the following statements are true', I think, or maybe 'what is the
only true statement in the following list.' It's been awhile since logic
classes though, maybe those are essentially the same as the original question,
I can't quite figure out what I think is wrong with the original question, it
just doesn't seem quite right to me.

Also, 4 is ill-stated -- "One of the above is true" \-- does that mean
_exactly_ one of the above is true, or _at least_ one of the above is true.
Although in that case I don't _think_ it matters because 1/2 are mutually
exclusive and 2/3 are mutually exclusive, and the combination of those two
means that 1/3 are mutually exclusive., so only exactly one of 1-3 can
possibly be true anyway.

------
PeterStuer
6 can not be true because if it where then 1,2,3,4 & 5 would be false, but
that would make 5 true as well which is a contradiction, therefore 6 can only
be false

1 Can not be true because we just determined 6 as false so 1 can only be false

3 can not be true because we just determined 1 as false so 3 can only be false

With 1 and 3 now proven false, 2 can only be true if 4 is false, but if 2 is
true then 4 would be true as well. This is a contradiction so 2 can not be
true.

With 1,2 and 3 now proven false, 4 is false.

5 is true since we just proved 1,2,3 and 4 to be false

~~~
thaumasiotes
Everyone in the comment thread seems to be focused on whether each proposition
can be true. This is looking for your keys under the lamppost despite the fact
that you dropped them in the dark. It's easy to determine whether a
proposition can be true.

But the question doesn't ask about that. It just asks "which of these is the
correct answer to this question?" There are no stated criteria for being the
correct answer, and there is no reason to assume that the correct answer must
be true or even capable of being true.

Consider this other question:

    
    
        Which of the following is the correct answer to this question?
          (A) This one.
          (B) This one.

~~~
zyx321
That's arguing about semantics. It seems obvious from context that the
intended interpretation of "the correct answer" is "the only true answer" —
since "true" and "correct" are nearly synonymous, and the use of definite
article singular implies that there exists exactly one answer.

~~~
gmfawcett
To be fair, making the argument that something is obvious from context is also
arguing about semantics. The other poster makes a defensible point (as do
you!).

------
northern-lights
I didn't see a logical explanation that explains why an option is correct or
not, so here's one:

Q Which answer in this list is the correct answer to this question?

1\. All of the below. 2\. None of the below. 3\. All of the above. 4\. One of
the above. 5\. None of the above. 6\. None of the above.

I'll assume "is the answer" to be "True" and "is not the answer" to be "False"
in my explanations to make it more readable.

Lets say 1 is True. This implies - 2, 3, 4, 5, 6 are also True. Now, if 2 is
True, then it means 3, 4, 5, 6 are False. This contradicts what 1 says. So, 1
cannot be True.

Lets say 2 is True. This implies - 3, 4, 5, 6 are False. If 3 is False, it
means both 1 and 2 cannot be True. Since, we've assumed 2 to be True, 1 can
indeed be False, so this is consistent. If 4 is False, then it means either
less than one or more than one out of 1, 2, 3 are True. Zero out of 1, 2, 3
cannot be True since it contradicts our assumption of 2 being True. So either
two or three out of 1, 2, 3 have to be True. Since, we've already established
than 1 is False, that leaves 3 to be True along with our assumption of 2 being
True. But 3 cannot be True since it contradicts 2. So 2 cannot be True.

Lets say 3 is True. This implies 1, 2 are True. We've already established that
1, 2 are False. Moreover, 2 being True will contradict 3 being True. So 3
cannot be True.

Lets say 4 is True. This implies that exactly one of 1, 2, 3 is True. We've
already established that they are False. So 4 cannot be True.

Lets say 5 is True. We now know that 1, 2, 3, 4 are not True. So, this is
consistent. We'll come back to this.

Lets say 6 is True. This implies that 5 is False. If 5 is False, it means all
of the above 5 - 1, 2 , 3, 4 are True which we've found not to be True. So 6
cannot be True.

That leaves only 5 to be answer to the question.

~~~
torstenvl
> _If 5 is False, it means all of the above 5 - 1, 2 , 3, 4 are True_

That is incorrect. "Not All False" does not imply "All True."

~~~
northern-lights
You're right, my bad. Negation of None of the Above is False is Some of the
Above is True. I will modify my explanation.

------
edejong
Solution in Scala, with JaCoP:

    
    
      import org.jacop.scala._
    
      object Main extends App with jacop {
        def forAll(l: Seq[BoolVar]) = l.foldLeft[BoolVar](true) { case (r, m) => r /\ m}
        def forAny(l: Seq[BoolVar]) = l.foldLeft[BoolVar](false) { case (r, m) => r \/ m}
      
        val v = List.tabulate(6)(i => new BoolVar(s"Answer ${i + 1}"))
        v(0) #= forAll(v.drop(1))
        v(1) #= ~forAny(v.drop(2))
        v(2) #= forAll(v.take(2))
        v(3) #= forAny(v.take(3))
        v(4) #= ~forAny(v.take(4))
        v(5) #= ~forAny(v.take(5))
        satisfy(search(v, input_order, indomain_min))
        print("Solution: " + v.filter(_.domain.contains(1)).map(_.id).mkString(" "))
      }
    

Results in:

    
    
      Solution: Answer 5

------
retsibsi
Maybe I made a mistake, but I think this can be solved really simply just by
taking each option in turn, assuming it is correct and looking for an
inconsistency.

1 can't be right because it both affirms and contradicts 2.

2 can't be right because it both denies and fulfils 4 (even if 'one' means
'one and only one', because we already know that 1 is false, and looking ahead
we can see that 3 is also false).

3 affirms 1, which we already know to be false.

4 affirms one of 1-3, which we have determined are all false.

5 is correct, as demonstrated by our finding that 1-4 are all false.

6 is false; we would know this even if we hadn't already worked out that 5 was
true, because 6 implies that 1-4 are all false, which implies that 5 is true.

~~~
jpxw
Exactly, I’m not sure why people are writing code to solve this. Just needs a
systematic approach.

------
novaRom
x is unknown, T=True, F=False, ttt = x or x or x, otherwise all connected with
"and"

By definition:

    
    
      #1: xTTTTT
      #2: xxFFFF
      #3: TTxxxx
      #4: tttxxx
      #5: FFFFxx
      #6: FFFFFx
    

Begin evaluation from 6.

Let's assume 6 is T so we have xxxxxT. In that case 5 need to be F. But if 5
is F, then any from [1,2,3,4] need to be T. But if any from [1,2,3,4] is T
then 6 cannot be T. Thus 6 cannot be T, so it is F if correct answer exists.

Now we have: xxxxxF

Next, we try to set 5 to T: xxxxTF. 1 cannot be T, 2 cannot be T, since their
last elements are different (TT!=TF, FF!=TF), 3 cannot be T because it
requires 1 and 2 be both T , 4 cannot be T because (1 or 2 or 3) are F.

So we have: FFFFTF

Thus only 5 is True.

------
sunstone
It seems all of the answers assume "You cannot assume there is only one
correct answer." However in "...answer in this list is the correct answer..."
to my mind the word "the" requires only one correct answer. If it was worded
'...a correct answer...' or '...one of the correct answers...' ok sure, there
can be more than one. But as it stands there can only be one correct answer so
the "correct" answers so far, are based on a false premise.

~~~
MrManatee
Regardless of the interpretation it's not a false premise. The uniqueness of
the answer is just an assumption that we don't need.

Suppose the question was instead: "What is the unique real number x for which
x^3 = 8?" We can guess and verify that x = 2 is a solution. And if we're
allowed to assume that this equation has a unique solution, then that's
enough. But if we work a little harder, then we don't actually need that
assumption: we can prove that x = 2 is the unique solution.

Similarly, in the original problem we could guess and verify that FFFFTF is a
solution. And if we're allowed to assume that the solution is unique, then
we're done. But again, if we work a little harder, then we don't need that
assumption: we can prove that FFFFTF really is the only solution.

------
eino
The question implies there is one correct answer.

Do you agree it can be rephrased as "Which answers in this list are the
correct answers to this question?", so as to make it more difficult, while
keeping the same answer?

~~~
brlewis
It could be rephrased that way without changing the answer, but I don't think
it would be any harder that way. You can still start with question 1 and
continue downward to logically deduce the answer.

------
bmer
Formal methods to the rescue:
[https://math.stackexchange.com/a/3437635/115703](https://math.stackexchange.com/a/3437635/115703)

------
dooglius
If you like this kind of puzzle, check out
[http://faculty.uml.edu/jpropp/srat-Q.txt](http://faculty.uml.edu/jpropp/srat-Q.txt)

------
Shinchy
I love that someone wrote a script to try and solve this problem.

------
tel
The question is phrased to mislead as this is a constraint satisfaction
problem over the boolean space 2^6. There may be 0, 1, or many correct
answers.

------
DantesKite
It's interesting that the StackExchange community explained the answer better
than the professor did in the original question.

------
slig
If you enjoy this kind of puzzle, look up the famous SRAT - self-referential
aptitude test.

------
fallingfrog
So tricky! Number 5 is the only one that works though I think

------
Derelicts
Very interesting, thanks for sharing.

------
mazsa
Only the 5th is true, Metamath: $( <MM> <PROOF_ASST> THEOREM=noneabove
LOC_AFTER=?

h50::noneabove.1 |- ( ph <-> ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) )

h51::noneabove.2 |- ( ps <-> ( -. ch /\ ( -. th /\ -. ta ) /\ -. et ) )

h52::noneabove.3 |- ( ch <-> ( ph /\ ps ) )

h53::noneabove.4 |- ( th <-> ( ph \/ ps \/ ch ) )

h54::noneabove.5 |- ( ta <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th ) )

h55::noneabove.6 |- ( et <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ ( -. th /\ -. ta
) ) )

56:50:simprbi |- ( ph -> ( ta /\ et ) ) 57:56:simprd |- ( ph -> et )
58:51:simp3bi |- ( ps -> -. et ) 59:57,58:anim12i |- ( ( ph /\ ps ) -> ( et /\
-. et ) ) 60::pm3.24 |- -. ( et /\ -. et ) 61:60,59:mto |- -. ( ph /\ ps )
62:61,52:mtbir |- -. ch 63:50:simplbi |- ( ph -> ( ps /\ ch /\ th ) )
64:63:simp2d |- ( ph -> ch ) 65:62,64:mto |- -. ph 66::3ioran |- ( -. ( ph \/
ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) ) 67:53:notbii |- ( -. th <-> -. (
ph \/ ps \/ ch ) ) 68:67,66:bitri |- ( -. th <-> ( -. ph /\ -. ps /\ -. ch ) )
69:68:anbi1i |- ( ( -. th /\ -. th ) <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ -.
th ) ) 70::pm4.24 |- ( -. th <-> ( -. th /\ -. th ) ) 71:69,70,54:3bitr4i |- (
-. th <-> ta ) 72::nbbn |- ( ( -. th <-> ta ) <-> -. ( th <-> ta ) )
73:71,72:mpbi |- -. ( th <-> ta ) 74::df-xor |- ( ( th \/_ ta ) <-> -. ( th
<-> ta ) ) 75:73,74:mpbir |- ( th \/_ ta ) 76::xoror |- ( ( th \/_ ta ) -> (
th \/ ta ) ) 77:75,76:ax-mp |- ( th \/ ta ) 78:55:simprbi |- ( et -> ( -. th
/\ -. ta ) ) 79::pm4.56 |- ( ( -. th /\ -. ta ) <-> -. ( th \/ ta ) )
80:78,79:sylib |- ( et -> -. ( th \/ ta ) ) 81:77,80:mt2 |- -. et
82:51:simp2bi |- ( ps -> ( -. th /\ -. ta ) ) 83::pm4.56 |- ( ( -. th /\ -. ta
) <-> -. ( th \/ ta ) ) 84:82,83:sylib |- ( ps -> -. ( th \/ ta ) )
85:77,84:mt2 |- -. ps 86:65,85,62:3pm3.2ni |- -. ( ph \/ ps \/ ch )
87:86,53:mtbir |- -. th 88:87,75:mtpxor |- ta 89:65,85,62:3pm3.2i |- ( -. ph
/\ -. ps /\ -. ch ) 90:89,87:pm3.2i |- ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th
)

qed:90,88,81:3pm3.2i |- ( ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th ) /\ ta /\ -.
et )

$= ( w3a simprbi mto wn wa mtbir wb sylib mt2 3pm3.2i pm3.24 simprd simp2d
simp3bi anim12i simplbi wo notbii 3ioran bitri anbi1i wxo w3o pm4.24 3bitr4i
nbbn mpbi df-xor mpbir xoror ax-mp simp2bi pm4.56 3pm3.2ni pm3.2i mtpxor )

ANZBNZCNZKZDNZOZEFNZVLVMVIVJVKACC
ABOZVPFVOOFUBAFBVOAEFABCDKZEFOZGLUCBVKVMENOZVOHUFUGMIPZABCDAVQVRG
UHUDMZBDEUIZDEUNZWBWCDEQNZVMEQWDVMVMOVNVMEVMVLVMVMABCUOZNVLDWEJUJ
ABCUKULUMVMUPUAUQDEURUSDEUTVAZDEVBVCZBVSWBNZBVKVSVOHVDDEVEZRSZVTT
DWEABCWAWJVTVFJPZVGDEWKWFVHFWBWGFVSWHFVLVSUELWIRST $. $)

~~~
gowld
Is this readable with more linebreaks?

~~~
mazsa
Not editable anymore but here you are (cf.
[http://us.metamath.org/metamath/set.mm](http://us.metamath.org/metamath/set.mm)
) :

h50::noneabove.1 |- ( ph <-> ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) )

h51::noneabove.2 |- ( ps <-> ( -. ch /\ ( -. th /\ -. ta ) /\ -. et ) )

h52::noneabove.3 |- ( ch <-> ( ph /\ ps ) )

h53::noneabove.4 |- ( th <-> ( ph \/ ps \/ ch ) )

h54::noneabove.5 |- ( ta <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th ) )

h55::noneabove.6 |- ( et <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ ( -. th /\ -. ta
) ) )

56:50:simprbi |- ( ph -> ( ta /\ et ) )

57:56:simprd |- ( ph -> et )

58:51:simp3bi |- ( ps -> -. et )

59:57,58:anim12i |- ( ( ph /\ ps ) -> ( et /\ -. et ) )

60::pm3.24 |- -. ( et /\ -. et )

61:60,59:mto |- -. ( ph /\ ps )

62:61,52:mtbir |- -. ch

63:50:simplbi |- ( ph -> ( ps /\ ch /\ th ) )

64:63:simp2d |- ( ph -> ch )

65:62,64:mto |- -. ph

66::3ioran |- ( -. ( ph \/ ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) )

67:53:notbii |- ( -. th <-> -. ( ph \/ ps \/ ch ) )

68:67,66:bitri |- ( -. th <-> ( -. ph /\ -. ps /\ -. ch ) )

69:68:anbi1i |- ( ( -. th /\ -. th ) <-> ( ( -. ph /\ -. ps /\ -. ch ) /\ -.
th ) )

70::pm4.24 |- ( -. th <-> ( -. th /\ -. th ) )

71:69,70,54:3bitr4i |- ( -. th <-> ta )

72::nbbn |- ( ( -. th <-> ta ) <-> -. ( th <-> ta ) )

73:71,72:mpbi |- -. ( th <-> ta )

74::df-xor |- ( ( th \/_ ta ) <-> -. ( th <-> ta ) )

75:73,74:mpbir |- ( th \/_ ta )

76::xoror |- ( ( th \/_ ta ) -> ( th \/ ta ) )

77:75,76:ax-mp |- ( th \/ ta )

78:55:simprbi |- ( et -> ( -. th /\ -. ta ) )

79::pm4.56 |- ( ( -. th /\ -. ta ) <-> -. ( th \/ ta ) )

80:78,79:sylib |- ( et -> -. ( th \/ ta ) )

81:77,80:mt2 |- -. et

82:51:simp2bi |- ( ps -> ( -. th /\ -. ta ) )

83::pm4.56 |- ( ( -. th /\ -. ta ) <-> -. ( th \/ ta ) )

84:82,83:sylib |- ( ps -> -. ( th \/ ta ) )

85:77,84:mt2 |- -. ps

86:65,85,62:3pm3.2ni |- -. ( ph \/ ps \/ ch )

87:86,53:mtbir |- -. th

88:87,75:mtpxor |- ta

89:65,85,62:3pm3.2i |- ( -. ph /\ -. ps /\ -. ch )

90:89,87:pm3.2i |- ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th )

qed:90,88,81:3pm3.2i |- ( ( ( -. ph /\ -. ps /\ -. ch ) /\ -. th ) /\ ta /\ -.
et )

------
cmonnow
answer -
[https://math.stackexchange.com/a/3437700/725808](https://math.stackexchange.com/a/3437700/725808)

The question displays a lack of understanding of language, especially the
relation between a word and its meaning. More specifically, the relation
between a noun and its pronoun.

Suppose Question 1 was:

> "What day is it today?"

And suppose Question 2 was:

> "Which answer in this list is the correct answer to this question? > Friday,
> Saturday, Sunday"

If someone asks, _' What question is the pronoun "this" in Question 2
referring to?'_

Then you can reply - _' Question 1'_

And then they can proceed to answer Question 2 as _' Saturday'_

Suppose I just ask you out of the blue : _' what's his height?'_

You will immediately respond : _' who are you talking about?'_

A pronoun must come AFTER a noun has been established. If it comes before, the
speaker must clarify the noun after.

In other words, you must be able to do a Ctrl-H Find & Replace in the original
sentence, and change "this" to whatever it's referring to, and the new
sentence should still make sense. That is the point of pronouns.

If I ask _' what is the weather in this city?'_

Does that question make sense without mentioning the city's name either before
or after? It might make sense grammatically, but practically, it is not
answerable.

If you now come back and say - Just replace the pronoun "this" with the text
of the question, then it becomes :

"Which answer in this list is the correct answer to "Which answer in this list
is the correct answer to this question" question?

Is the question answerable now ? Still NO. Because there is still one
unresolved "this". Ad-infinitum.

So, to answer your question:

> 'which question ?'

The question, as is, is not answerable. Because it does not make sense until
you resolve what 'this' refers to. Until then, it is just a bunch of words
without a corresponding meaning. It's not a paradox or a contradiction.
Barber's 'paradox', 'This sentence is false' etc. all are basically just a
poor understanding of language/pronouns.

You might as well ask :

> 'Which answer in this list is the correct answer to oogabooga question?'

------
pvaldes
what is the question?

5 and 6 are the only possible. But if 5 is true then 6 is false

None of the above can be true only with "more than one of the above (but
different than all, none or one)", but then we would not be talking about
_the_ correct answer. It is assumed than only an answer can be correct.

I suppose than 6 then.

Updated: you are right, 6 can't be true, is the 5

~~~
nmeofthestate
I got as far as your first sentence - "what is the question".

I can't work out what the question is asking, so I don't know how to work out
what answers are correct/incorrect.

