The following is the unexpected hanging paradox:
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.
Observations
In my opinion the above reasoning is a case of a "proof by assumption" with extra steps. Here's why:
- prisoner assumes hanging didn't occur on Monday, ..., Thursday to conclude that it can't happen on Friday
- he works backwards from Thursday to Monday using the same argument as for Friday
The second step is redundant since we already established in step 1, that hanging did not occur on Mon, ..., Thu.
In this way, the "paradox" is reduced to a single implication: (no hanging Mon, ..., Thu) -> (no hanging on Fri). It's easy to agree with and not a paradox.
Is my thinking correct here?