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The Surprise Execution Paradox

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My roommate in college told me this one:

A man in the old west is found guilty of a crime. At his sentencing on a Sunday, the honest judge tells him this: "You are found guilty and are sentenced to death by hanging. You will be hung at high noon one day this week. However, the day of your hanging will be a complete surprise to you, and you will not know if it is the day until the morning of that day."

Later in council, his attorney tells him excitedly, "You've gotten off! There is no way the sentence can be carried out and the honest judge keep his word! You see, there is no way you can be hung on Saturday, because it's the last day of the week, and you'll know Friday night that your hanging will happen the following day, that makes Friday the last day you can be hung. But that can't be, because then you'll know when Thursday night rolls around that you'll be hung on Friday, because its the last day. But then you'll know Thursday, so you cannot be hung Friday, making Thursday the last day, and so on until today!"

The man, having heard this, confidently strolls to his jail cell having just avoided a death sentence. But when he is pulled out of his cell Tuesday morning to be hung at noon, he is 100% surprised!

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Posted · Report post

could it be that the noon the judge is talking about is noon somewhere else, after all it is high noon somewhere all the time. (or it could be midnight)

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Posted · Report post

At his sentencing on a Sunday, the honest judge tells him this: "You are found guilty and are sentenced to death by hanging. You will be hung at high noon one day this week. However, the day of your hanging will be a complete surprise to you, and you will not know if it is the day until the morning of that day."

Later in council, his attorney tells him excitedly, "You've gotten off! There is no way the sentence can be carried out and the honest judge keep his word! You see, there is no way you can be hung on Saturday, because it's the last day of the week, and you'll know Friday night that your hanging will happen the following day, that makes Friday the last day you can be hung. But that can't be, because then you'll know when Thursday night rolls around that you'll be hung on Friday, because its the last day. But then you'll know Thursday, so you cannot be hung Friday, making Thursday the last day, and so on until today!"

The man, having heard this, confidently strolls to his jail cell having just avoided a death sentence. But when he is pulled out of his cell Tuesday morning to be hung at noon, he is 100% surprised!

The problem is there is a lawyer trying to use logic.

He is right, but only to a point. Yes, if he had not been hung by Saturday, then it would not be a surprise on Saturday morning as that was the only day left. The judge started off the sentence telling him he would be hung one day this week, so the fact he is being hung isn't a surprise -- only the day. So if you rule out Sunday (the day of the sentence) and Saturday (for reasons noted above), there are five possible days. On Monday, he has a 20% chance of being hung. If that doesn't happen, then on Tuesday, he has a 25% chance of being hung. On Wednesday, 33.3%, On Thursday, 50%, and if lasts till Friday, 100%.

The last day he always has 100% chance. So if you try to keep backing the day up using the lawyer's logic, you end up with Monday being 100%.

If you are using the logic that the judge is honest in saying it will be a surprise, you must also use the logic that he is honest in saying it will happen one day this week.

And ... he was surprised, right? And he was hung, right? So the judge was right.

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Posted · Report post

I first saw this in a book by Martin Gardner called (I think) Mathematical Puzzles and Paradoxes.

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Posted · Report post

so did i. Martner Gardner explained the solution to the faulty logic, too. The lawyer reasoned wrong because you can't even be sure if the VERY LAST DAY is the hanging day, even if all days before it have gone by and you havent been hung. It's explained with this:

There is a husband and a wife. The wife's birthday is tomorrow. Her husband says: "For your birthday tomorrow I'm going to get you an unexpected gift. You'll have no idea what it is until you open it... It will be that gold necklace we saw in the jewelry store."

The wife knows her husband to be truthful, but this confused her. The more she thinks about it, the more confused she is. Her husband made two statements there: the gift will be unexpected, AND the gift will be the golden necklace. Since both couldnt be true her husband isnt being truthful, he has to be lying. So he might get her the golden necklace. Or he might get her that new vacuum cleaner they need. Who knows. She has no idea.

The next day she opens her gift and its the golden necklace from the jewelry store. It is both unexpected AND the golden necklace. Her husband was being truthful all along, and he knew it. She didn't until she opened it though.

In this way even if the last day possible for the hanging is tomorrow, we cant be sure if it will happen.

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Posted · Report post

Putting it another way, the more certain the man is
[on Friday night, should he live that long] that he
cannot be hanged on Saturday [the last possible
day for the hanging] the more surprised he is
when it happens. It's possible because of the
prisoner's unwavering belief in the truth of the judge's
statements.

As with other semantic paradoxes, the assumption
that what is being asserted is true comes into play.
If the prisoner didn't believe what the judge said
was true, he might [expectantly] dread a Saturday
hanging, thereby precluding it!

The moral:
Never believe a judge. It might cost your head.

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Posted · Report post

The problem is there is a lawyer trying to use logic.

He is right, but only to a point. Yes, if he had not been hung by Saturday, then it would not be a surprise on Saturday morning as that was the only day left. The judge started off the sentence telling him he would be hung one day this week, so the fact he is being hung isn't a surprise -- only the day. So if you rule out Sunday (the day of the sentence) and Saturday (for reasons noted above), there are five possible days. On Monday, he has a 20% chance of being hung. If that doesn't happen, then on Tuesday, he has a 25% chance of being hung. On Wednesday, 33.3%, On Thursday, 50%, and if lasts till Friday, 100%.

The last day he always has 100% chance. So if you try to keep backing the day up using the lawyer's logic, you end up with Monday being 100%.

If you are using the logic that the judge is honest in saying it will be a surprise, you must also use the logic that he is honest in saying it will happen one day this week.

And ... he was surprised, right? And he was hung, right? So the judge was right.

I would guess you aren't as blonde as you are sarcastic . Don't take that the wrong way, though. It's perfect.

I wouldn't be too sure about the maths in your assessment of the situation, although I can't see a way of disproving it. But the situation is simply that he WAS hung some day that week, as the Judge promised, and it WAS a surprise to him, also like the Judge said...

Actually, what the lawyer said also made good sense. It was all perfectly logical. BUT that only gave the man the certainty he would not hang, and THAT made the surprise complete. After all, he was convinced he would not hang, but he did!

What else is there? An eternity in hell for this man

BoilingOil

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Posted · Report post

Pitiful attempts at off-color humor notwithstanding (and there are always some), people who are executed by hanging are hanged, not hung. The judge may have been honest, but his grammar was substandard.

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Posted · Report post

Huh, I think i just replyed the the same question yeserday, only about a math teacher and student predicting a drop quiz. I don't think this is paradox really, yet to some extent it is - you could say that the guy's lawers resoning is. Oh, here it is! - that one about the math teacher...:

Predicting a Drop Quiz

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Posted · Report post

Maybe the judge forsaw the flawed logic in his own ruling. He predicted the reactions of the condemed man and his lawyer alike. He gave them a sure outcome in thier favor using deductive reasoning. A false sense of security that only reason could secure. He wanted to give them a concrete belief that the condemed man's execution wasnt logically possible therfore making his actual execution a complete suprise. If the condemed man thought that he could actually be exucuted sometime during the week then he would be expecting the act to occur and would in at least some way be unsuprised. The judge used logic to predict the best way to carry out the sentence. The judge demonstrates how a persons own logic can be there downfall.

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Posted (edited) · Report post

Sorry for bumping such an old thread, but it's a good one and the answers haven't explored every aspect.

The absolutely honest judge has given the prisoner two contradicting statements: You will be surprised, and there is no day that you can be executed that you can, in fact, be surprised.

Similarly the absolutely honest husband who makes two condradicting statements to his wife, I'm giving you a gift tomorrow that will surprise you, and it will be that gold necklace we saw at the jewelry store.

In both cases this is a paradox only if the future is perfectly predictable, which it is not. There could be a prison riot and the prisoner could escape. The husband could develop amnesia and get the wrong present.

This operates on the same premise as another old thread here in the paradox section: The Minority Report Paradox. If the seers perfectly predict the future they will know that the crime they are supposedly seeing will be prevented because of their prediction, so there will be no crime for them to foresee. It is a paradox only if they can actually predict the future perfectly; however the physics tells us that it is intrinsically (by the very nature of the laws of physics) impossible for a perfect prediction of the future because, if nothing else, random locations and motions of sub-atomic particles do not have precise values, so cannot be perfectly observed.

Mathematics tries to cram a false absolutism upon an arbitrary, random world. Perfect logic is fundamentally flawed. Here we have the real paradox :P

Edited by seeksit
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Posted · Report post

Putting it another way, the more certain the man is

[on Friday night, should he live that long] that he

cannot be hanged on Saturday [the last possible

day for the hanging] the more surprised he is

when it happens. It's possible because of the

prisoner's unwavering belief in the truth of the judge's

statements.

As with other semantic paradoxes, the assumption

that what is being asserted is true comes into play.

If the prisoner didn't believe what the judge said

was true, he might [expectantly] dread a Saturday

hanging, thereby precluding it!

The moral:

Never believe a judge. It might cost your head.

I agree, In this case the judge told him he will be surprised. The prisoner(with help from his "logical" lawyer) try their best at disproving the fact that they will be surprised or even hung at all. So it makes sense he would be surprised, after being led to believe that there was no way the judge could hold up to his promises.

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Posted · Report post

...On Monday, he has a 20% chance of being hung. If that doesn't happen, then on Tuesday, he has a 25% chance of being hung. On Wednesday, 33.3%, On Thursday, 50%, and if lasts till Friday, 100%.

The last day he always has 100% chance...

Isnt this incorrect. its like the 2 goat and a car behind the doors game show thingy. it is always a 20% chance of happening. on the last day he KNOWS it is a 100% chance that he will be hung that day, but there is still a 20% chance of being hung on friday....

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The problem with these types of paradoxes is that the attourney (in this case), utilizes mathematical logic and not everyday common sense. The only true day that would not be a surprise is saturday. But this day would not be a surprise, only by friday night. On Thursday night both Friday and Saturday are free game. One can't just assume that the judge would eliminate the last day just because it is the last.

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Posted · Report post

The judge is either divine, psychic, or in cohorts with the attorney. Such a statement had to be made to put the criminal off guard.

There is a solution, given I play the role of the criminal this time. After my attorney makes his statement, I would then say "Your honor, I choose not to follow the logic of my attorney as it surely condemns me." Then on any day I am summoned to hang, I then say "I am not at all surprised." to the one in charge. If the judges orders are then followed properly, I will not hang as a result of action.

Arseuming does more than make an arse out of you and me, it can also be fatal :)

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Did the lawyer forget that there is kind of more days then Thursday, Friday, Saturday and Sunday???

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