The Hangman's Paradox

Unlike other paradoxes I've heard, I've never been able to resolve this one, and it's always intrigued me. It's also known as the Paradox of Prediction, the Unexpected Exam Paradox, and a few other names. I'll give one version of it.

You are in prison, and sentenced to death. You are told that you will be hanged on an unexpected, 'surprise' day. You are informed that it will be any day from Monday to Friday (either Mon, Tues, Wed, Thurs, or Fri), but you are not told which day.

What is the one day on which the hanging would not be a surprise? Think about it for a moment and the answer is obviously Friday. If you know that you're going to be hanged by Friday at the latest, and you are still alive on Friday, you know you will be hanged today. When the executioners come for you, you will not be surprised.

So, you cannot receive your 'surprise' hanging on Friday. It would not be a surprise. So, you know that your surprise hanging will happen from Monday to Thursday. Very well. But now we have the same problem with Thursday that we had with Friday. Come Thursday, you know that this is the last day you can receive your surprise hanging, and so you know you will be hanged today, and it will not be a surprise.

Now you know that the surprise hanging can only happen from Mon - Wed. The pattern repeats until there is no day left on which you can be genuinely surprised.

But, in real life, we know that you can be surprised with the hanging on one of the days, even though it's been logically deduced that you can't. How does this make sense?

Very nice:)
Ok so i understand it like this:
The range of the suprice days differs depending on the actual day.
On monday the range is the biggest - <mon , fri> possibility that it will happen today is 1/5
On tuesday - <tues , fri> - possibility 1/4
and so on...
<wed , fri> - possibility 1/3
<thurs, fri> - possibilty 1/2
<fri, fri> - possibility 1 - so only here there will be no suprise :)

I don't think you understand the paradox. Either that or I don't understand your response.

Since on <fri, fri> the probability is 1, it will not be a surprise on Friday. Since a requirement of the hanging is that you will be surprised, you know in advance that it will not happen on Friday. So, the probability of being hanged today when it'sThursday is also 1. And the pattern repeats, so the probability of being hanged today when it's Wednesday is also 1, and I hope you see the problem now.

Comic form!

http://trollscience.com/image/f/full...738a50658d.jpg

The exact problem with the paradox is a bit sticky. I think the root of it is that at each stage you are saying 'you must punish me today therefore you cannot punish me today'; it's contradictory self-reference. As soon as you've established 'you cannot punish me today', the very argument supporting this fact fails, because the punishment happening today would now indeed be a surprise.

A few thoughts and possible arguments:

1. The paradox does not actually make sense. If you're told you're going to be executed at noon on one of five days this week, then you could easily say that it's impossible to be surprised by the guard turning up on a certain day, because you knew it was very likely. It's like tossing a fair coin that can be either heads or tales, and being surprised if it turns up heads.

2. The paradox is self contradicting, because as Xei states, as soon as you make any claims like "you can't execute me today because I'm expecting it" executing the person on that day will indeed be a surprise.

3. The paradox moves the goalposts. If told straight away after the above sentence that you'll be executed on Friday then it would indeed be a surprise because the odds are, all things being equal, that the other 4 days would be more likely. Getting to Friday and claiming it is no longer a surprise through the process of elimination shifts the goalposts by taking advantage of the knowledge gained during the week.



One possible method of escaping though, according to the paradox, is if the accused sets his own expectations, and believes he will be executed on a given day, thus not being surprised when the guard turns up, and being surprised if he survives. It's an irrational declaration, but as long as he's able to sincerely believe it, it wouldn't matter.

I don't really understand it. I've also read the entry on Wiki. Yes, I understand the basis of the argument. If he hasn't been hung by Thursday, then it won't be a surprise on Friday, and he can't receive his surprise hanging. Well, why don't they just hang him Tuesday instead? Then it's a surprise. As long as the surprise is set between Monday and Friday, but the actual hanging is between Monday and Thursday ("startdate and enddate-1"), there's no issue, the hanging is a surprise. Surprise = element of chance. Because the judge introduced the word 'surprise', the hanging must be between day 1/5 and day 4/5. If the word 'surprise' was omitted, it could be from day 1/5 to 5/5 where on day 5/5 there is no surprise.

I don't really see how the argument re the lack of surprise he would have received had he not been hung by noon on Thursday some how carrying back in time and meaning that no day will be a surprise therefore he can't get hung?

Why is this "Even though it is apparently simple, the paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.[3]"? Perhaps I'm missing something..?

hmm maybe i dont get something right but practicly... lets asume that it is thursday and the man is still alive :) If they hang him today it will sure be a suprise ( cause they can do it today or tomorow). I dont see how the probability could be 1. No paradox in my opinion. Ok it coud be a paradox if the time woludnt be linear but its not.
btw the comix is terrible :P

PS thinking out of the box : it would be the most suprizing to be sentenced on friday dont u think ? ( because its the least You expect :) )

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Originally Posted by nrg

it could be a paradox if the time wasn't linear but it is.

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Fixed. and
This.

I like these topics, they make my head hurt :)

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Originally Posted by Xei

I think the root of it is that at each stage you are saying 'you must punish me today therefore you cannot punish me today'; it's contradictory self-reference. As soon as you've established 'you cannot punish me today', the very argument supporting this fact fails, because the punishment happening today would now indeed be a surprise.

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You might be right.

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Originally Posted by Photolysis

1. The paradox does not actually make sense. If you're told you're going to be executed at noon on one of five days this week, then you could easily say that it's impossible to be surprised by the guard turning up on a certain day, because you knew it was very likely. It's like tossing a fair coin that can be either heads or tales, and being surprised if it turns up heads.

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Unless by 'surprised' what is meant is that the probability of being executed today is less than 1.

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Originally Posted by Photolysis

2. The paradox is self contradicting, because as Xei states, as soon as you make any claims like "you can't execute me today because I'm expecting it" executing the person on that day will indeed be a surprise.

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I'm not saying this is invalid, it's probably somewhere near the root of the problem. But, in real life, if it's Friday and you're still alive, the conclusion isn't that you've escaped your surprise hanging. The conclusion is that you're going to be hung today, but it isn't going to be a surprise. And the paradox states that it's impossible to be surprised when you are hanged. In real life, it's true that on Friday you wouldn't be surprised. Does the same even hold true for thursday? It seems like it does. Wednesday? It seems like it doesn't, but I'm not sure why.

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Originally Posted by Photolysis

3. The paradox moves the goalposts. If told straight away after the above sentence that you'll be executed on Friday then it would indeed be a surprise because the odds are, all things being equal, that the other 4 days would be more likely. Getting to Friday and claiming it is no longer a surprise through the process of elimination shifts the goalposts by taking advantage of the knowledge gained during the week.

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I don't see how the goalposts are being shifted. There's no problem with taking advantage of the knowledge gained during the week when the hypothetical scenario involves already having lived through that week.

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Originally Posted by apsinvo

I don't really see how the argument re the lack of surprise he would have received had he not been hung by noon on Thursday some how carrying back in time and meaning that no day will be a surprise therefore he can't get hung?

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What if he isn't hung by noon on Wednesday? In real life, he would know that he'll be hung tomorrow with 100% certainty, on Thursday. Because he knows that the judge wouldn't wait until Friday, since then it wouldn't be a surprise. I don't know if that clarifies or confuses it.

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Originally Posted by nrg

lets asume that it is thursday and the man is still alive :) If they hang him today it will sure be a suprise ( cause they can do it today or tomorow). I dont see how the probability could be 1. No paradox in my opinion.

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No, on thursday it genuinely wouldn't be a surprise, not even if the situation were real. He knows that the judge wouldn't choose to hang him on Friday, since then it would not be a surprise. So, if it's Thursday, he knows his hanging is coming today. The same goes for Wednesday. For the reason above, he knows the judge cannot wait until Thursday, so on Wednesday he should know that he'll be hanged today.

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Originally Posted by nrg

PS thinking out of the box : it would be the most suprizing to be sentenced on friday dont u think ? ( because its the least You expect :) )

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In real life, if it's Friday and you aren't hanged yet, you know that you'll be hanged today, so it wouldn't be a surprise. You'd conclude that the judge simply made a mistake and did not surprise you. The paradox states that it's impossible to be surprised, not that it's impossible to be hanged. I think that's taking it too far. That's the problem I have with the the story in the comic posted, with that way of presenting the paradox.


The thought has occurred to me to simplify the issue. What if the judge told you that you are to be hanged either today or tomorrow, but it will be a surprise? Realistically, I think it is truly impossible to be surprised in this situation. The paradox works. If 'tomorrow' is reached and you haven't been executed yet, you know you'll be executed today and it will not be a surprise. So on the first day, realizing the above, you know you're going to be hanged today, and so it also won't be a surprise.

What if there are three days? The judge tells you that you will be executed either Wednesday, Thursday or Friday. That makes it a bit difficult. Do you know that you're going to be hanged on Wednesday, and therefore is it also impossible to be surprised?

Maybe the core of the issue is that people simply aren't going to think ahead. In the three-days situation, if the scenario were completely real, I would conclude that I would probably be hanged on either Wednesday or Thursday, without knowing which day. Although I think the judge would be likely to realize he can't sentence me on Friday and keep the surprise, and therefore not sentence me on Friday, I don't think he's going to have the sense to think ahead and realize he cannot sentence me on Thursday either for the same reason.

All you have to do is lie and hang him on saturday.

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Originally Posted by Dianeva

I'm not saying this is invalid, it's probably somewhere near the root of the problem. But, in real life, if it's Friday and you're still alive, the conclusion isn't that you've escaped your surprise hanging. The conclusion is that you're going to be hung today, but it isn't going to be a surprise. And the paradox states that it's impossible to be surprised when you are hanged. In real life, it's true that on Friday you wouldn't be surprised. Does the same even hold true for thursday? It seems like it does. Wednesday? It seems like it doesn't, but I'm not sure why.

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Because Friday opens back up as a possibility when you move your consideration from Thursday to Wednesday

This paradox is based on the same set of principals as one that I thought of once.
Statistics show us that a quarter will land on tails 50% of the time. Therefore, flipping the quarter twenty times should result in the quarter landing on tails five times.
At this moment the quarter has landed on heads five times in a row. Statistically speaking this means that there is a 100% possibility that the next time the quarter is flipped it will land on tails. However, logic tells us that it is STILL a 50/50 shot, and that EACH TIME the quarter is flipped there is STILL a fifty percent chance that it will land on tails, even though this would violate the fundamental and overall statistic.

-shoves the quarter in a coke machine-

I don't see how that relates to this paradox. That isn't even a paradox. Statistically, the expected number of Heads landed on in 10 coin clips is 5. But there's nothing in statistics saying that it has to. All the 'expected value' is, is the number we'd most likely expect to get (that's not the exact definition but it's close enough for this discussion) before we've begun the coin tossing experiment. Statistically speaking there is not a 100% possibility that the next time it's flipped it will land on tails, no matter how many heads in a row we've gotten. Assuming that there's anything higher than a 50% chance of getting tails on a fair coin would be the Gambler's Fallacy.

Edit: I just noticed you mentioned flipping the coin 20 times and expecting it to land on tails 5 times, when you obviously meant 20 and 10, or 10 and 5. Maybe you confused yourself with the word 'quarter', lol.

The only time it would not be a surprise is if he was hung on friday, which means the judge (or whoever told him this) lied to him about it being a surprise. Think about this: if he was telling the truth, you could tell that it would not be on friday, but since you know that, the same principal applies if it was on thursday. Thursday would come, and, since the judge said it would be a surprise and you know this paradox, you would know that it would not be a surprise on friday, so it must be today (thursday). But wait a minute, then it wouldn't be a surprise again even though it's on thursday! So then on wednesday you know it would not be friday, but now you know it would be today (wednesday) because it would not be a surprise on friday nor thursday. But then, its not a surprise again! This would keep occurring until it would be monday, and it would not be a surprise because you knew it would be tonight. Therefore, according to this paradox, it is impossible to be surprised when given a date of something ocurring between X and Y day.

However, you would then have the original 5 days to guess from, therefore it seems to be able to be a surprise. This paradox is simply contradicting itself into saying something can't be a surprised, but when analyzed from a different angle, it can be a surprise.

I SHALL COIN THE TERM, THE STARBURST PARADOX! (You heard it here first)

(Good luck understanding what I'm trying to get across ;) )

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Originally Posted by IndieAnthias

Because Friday opens back up as a possibility when you move your consideration from Thursday to Wednesday

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this

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Statistically speaking this means that there is a 100% possibility that the next time the quarter is flipped it will land on tails.

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No there isn't. Statistically AND logically it's 50-50 either way. This is basic probability here...

Statistically speaking, the most likely final result from the position of getting 5 heads from the first 5 tosses would be 8H-2T or 7H-3T. Probability only gives an indication of the likelihood of outcomes in advance. Past results have no bearing whatsoever on future outcomes because the system has no memory.

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even though this would violate the fundamental and overall statistic.

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No it wouldn't, because the overall statistics are based around an infinite number of events, and cannot be 'violated' by a finite chain of events.

sloth, you really need to learn about subjects before you speak of them...seriously.

The reason why its a paradox is because artificial words have created an artificial requirement that doesn't exist in reality. You can never guarantee when someone is or isn't surprised. Therefore being surprised can never be a requirement for being hanged. Only death D=

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Originally Posted by juroara

The reason why its a paradox is because artificial words have created an artificial requirement that doesn't exist in reality. You can never guarantee when someone is or isn't surprised. Therefore being surprised can never be a requirement for being hanged. Only death D=

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A paradox is when both contradictory answers both have beneficial outcomes so that choosing either would be good. Therefore meaning that two contradictory answers can both be correct.

To the OP:

I see the exercise you are giving. Basically if one were to assume that everyday they were going to be hanged (Mon-Fri) then there can never be a day in which you are surprised about being hanged yet if a day were ever to come when you would actually be surprised because your mentality was that you would never be surprised on the day. I see the paradox in it. One side is that you should not be surprised because you knew you were going to be hanged between Mon-Fri yet on the other side you should be surprised because you assumed that if you weren't surprised.

Yup that's a good old fashioned paradox.