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It is a logic puzzle left by my discrete math professor. We are learning propositional logic now. Here is the puzzle:

Among all the students in our class, at most two students give the 'correct' answer. There may be multiple 'correct' answers, even if someone gives the 'correct' answer, he or she may not be right ultimately. So the question is:

What is the 'correct' answer?

Note that this question is about marking exam papers so the teacher should mark the papers in a usual, normal and real-world way, i.e. answers like randomly choosing zero or one or two students' answer as the ultimate right answer(s) are not the initial aim of the question. So before posting your answer, think twice that in real world, do teachers mark test papers in that way you suggested?

For example, I tried like this:

The 'correct' answer is: "My answer and at most one answer of all the other students are right."

First my logic is right, explanation: If no one else or someone else give the same answer as me, then as my correct answer states, we are both right; while if more than one students give the same answer as me then the answer statement itself contradicts the fact so all our answers are wrong, i.e. even if all these answers are 'correct' yet still no one is right.

But my professor said that this is still kind of weird as in real world, teachers don't take notes of the number of 'correct' answers while marking and then decide who is/are right. And he gave me a hint that the answer is about paradox. But I can't figure it out. So in your opinion, what is the 'correct' answer (not limited by the hint paradox)?

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  • $\begingroup$ It is unclear what you are asking here. $\endgroup$ – Matsmath Sep 24 '16 at 9:51
  • $\begingroup$ @Matsmath add some explanations so can you understand it now? $\endgroup$ – pjpj Sep 24 '16 at 9:57
  • $\begingroup$ What does it mean to "mark papers in a usual, normal and real-word way"? I have no idea what you are referring to. $\endgroup$ – celtschk Sep 24 '16 at 13:27
  • $\begingroup$ @celtschk That means the teacher judge a paper's correctness only by the content of the answer (objectively), counter examples such as casually select some papers' answers to be correct, or judge one paper after comparing its answer with other papers, etc. Just want to stress that it is a common procedure of marking papers in order to avoid those brain teaser answers. $\endgroup$ – pjpj Sep 24 '16 at 13:55
  • $\begingroup$ I am just guessing here in the dark, but maybe the question itself is important, to understand what kind of paradox lurks behind the correct answer? $\endgroup$ – Matsmath Sep 24 '16 at 14:20
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Hopefully this is submitted in time to be correct.

“Fewer than 2 answers to this puzzle have been assessed as correct.”

If this is the first correct answer to be assessed (graded or posted)...

...it is correct immediately by default.

When the next answer identical to this comes along...

...it might get stuck in a paradox, whose resolution could create at most one more correct answer, which would amount to 2 correct answers that have been assessed, which would make this answer incorrect retroactively but not change the fact that it was assessed as correct.

Note that this answer is not limited to a point in time, only to the facts at the present, like saying “it is daytime.”

Regardless...

After the next answer like this, no further answers will be assessed as being correct so that there is no way that more than 2 will be assessed as correct.

Better post this before really having a chance to think it through or even proofread. Wouldn't want to be the one stuck in a paradox.

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  • $\begingroup$ I think your logic is right, but still there is one thing in the question statement that 'There may be multiple correct answers' which I think might be there may be multiple different correct answers. How about that? $\endgroup$ – pjpj Sep 24 '16 at 10:16
  • $\begingroup$ No time to reply, Too busy preparing a second answer. (Actually, this answer was mainly meant as a concrete test of the puzzle's parameters. I'll gladly delete it if it's off course, and gladly leave it if it servers as a bad example.) $\endgroup$ – humn Sep 24 '16 at 10:23
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Real world?

If my answer is right I'll give you $10.

Reasoning:

If more than 2 students give this as an answer you're going to get caught, and while the answer was 'correct', you most certainly won't be "right" ultimately.

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  • $\begingroup$ Man, LMAO, but why more than two then will be caught... $\endgroup$ – pjpj Sep 24 '16 at 15:15
  • $\begingroup$ Because I can NOT picture three students knowing something like that without all of them knowing, which then means the administration also knows. $\endgroup$ – Dark Matter Sep 24 '16 at 17:26
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You are supposed to guess that...

There is no 'correct' answer.

Among all the students in our class, at most two students give the 'correct' answer:

As there is no 'correct' answer exactly 0 students will give the 'correct' one, and 0<=2.

There may be multiple 'correct' answers, even if someone gives the 'correct' answer, he or she may not be right ultimately:

This is not contradicted.

The notation also needs to be fair:

To get the point you need to answer the meta-question "what is the 'correct' answer?" Not to actually give this answer. Each student may come up with this reasoning and all get the points independently of what the others would answer.

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  • $\begingroup$ This seems to work and it is paradoxical... +1 $\endgroup$ – Brent Hackers Jan 26 '18 at 17:16
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How about

"Exactly two of us have this right"

That way

Either two of you give this answer and are correct or, paradoxically(?), more than two do (There may be multiple 'correct' answers) and are therefore wrong (may not be right ultimately) even though the answer is 'correct'? ...But this is effectively the same as your answer absent its conditional nature.

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  • $\begingroup$ Or "at most two ...". $\endgroup$ – Lawrence Jan 26 '18 at 16:02

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