system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: The Unexpected Hanging paradox.

I understand most of the paradox except one bit, how the hell does the prisoner eliminate the days of his hanging? I couldn't follow his logic at all.
gpt
First, (for the lazy) I'll paste the paradox here

>A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

> Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

> He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

> The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

Think about this with just  Friday.  You are given the conditions for your hanging.  Noontime, knock on the door - you will die.  Now it's a process of elimination.  If Thursday noon has passed and nobody knocked on your door, then you *will* be hanged on Friday.  Therefore, that cannot be a surprise for you.  This eliminates Friday as there is no element of surprise.  And because there is no element of surprise, it doesn't meet the judge's requirements (*the execution will be a surprise*).

Taking this a step further, since Friday cannot be the day of hanging, *the last possible day to hang is Thursday*.  But then if Wednesday afternoon goes by and there's no knock, it's pretty obvious he'll hang on Thursday.  That means Thursday isn't a surprise either.  Fails the judge's requirements.  No hanging!  It needs to be a surprise!

Again.  Since Friday and Thursday are out, clearly the last day for a surprise is Wednesday.  But if Tuesday goes by and no knock, then hanging on Wednesday is no surprise.  Clearly it can't be Wednesday!

Similarly, Tuesday is eliminated.  And Monday is eliminated.  All because of that clause (*the execution will be a surprise*)


It's because of this conclusion that the knock on Wednesday is a surprise.  If he had not come to the above conclusion, it wouldn't have been a surprise.  So the surprise is a self-referential one.   It's quite meta.