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.
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.
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:
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:
But is it an answer to the question? If the question were something completely different, "How green is interstellar space?", then we would still find 5 as the answer. But "None of the above" is a non-answer. And "What is the answer to this question?" is not well defined.
I wish this wasn't the top comment, and not because it is a bad comment, but because I caught a peak of it before I could try the puzzle without hints.
In my opinion there isn't an answer to the question.
The italicized _this_ is what's important. You need to answer the question itself. As far as I can tell, everyone is caught up reading the answers as
1. All of the below _are true_.
2. None of the below _are true_.
etc.
but the answers don't say that. They don't refer to other answers. The are direct responses to the question. None of those answers make sense in context.
"Which answer in this list is the correct answer to this question? All of the above." That doesn't make sense.
"All of the above _are true_" (etc.) would make sense, but as stated none of the answers provided offer a coherent response to the question.
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?'
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.