The One-Day Unexpected Hanging

Here’s one of the most famous of all paradoxes:

A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any other day. Nevertheless, the executioner unexpectedly arrives on some day other than Friday, surprising the prisoner.

(Source: Mathworld.)

There are various arguments resolving the paradox (with no consensus), including one that claims the paradox still applies when there are only two days.

I have what may be a slight variation. With a problem this popular, it’s hard to be original, so I doubt it, but —

Let’s suppose it’s Friday, and the prisoner is told they will be hung today. They’re also told it will be unexpected which day they will be hung.

The prisoner detects an immediate contradiction, and so concludes he won’t be hung. He has an unpleasant surprise when he’s hung anyway.

The essential part here seems to be the “paradox” condition of the prisoner’s reasoning, which works like an extra day. In other words, approaching the problem with only the values “true” or “false” is not enough.

6 Responses

1. For me, the answer is actually that the guard got it wrong / lied. I know it’s tradition in these problems that the questioner is infalliable, but in this case it just doesn’t work.

2. This “paradox” is typical of most that I have seen. The element of confusion or conflict which prevents resolution is “unknowingly” injected by the listener who grafts the error onto the problem and thereby creates his own prison of irreconcilable reason. Why, because: 1) it was done subtlety; without conscious awareness, and; 2) as it is a creation of the problem solver, and indistinguishable from him, and being part and parcel of him it cannot be identified to be excised and removed from its place of importance; that of confounding the victim.

The error that is injected by the listener is the strict limitation on the meaning of “hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens”. Does the paradox have life (pardon the pun) if the listener heard instead the equally accurate statement: “hangings are only done Monday through Friday and the condemned only informed the day of the event.” Of course it requires retraining and damaging your brain as you progress through law school to understand this.

3. I’m not entirely buying the lying angle, partly because this was originally based on a real problem. (It had to do with a drill during WWII, if I remember right.) There was no malice on the part of the speaker: they really did mean to have a drill that week, and it really was a surprise which day.

One could argue the language of the word “surprise” needs to be more precisely formulated, but it has: define surprise as “able to deduce the day of” and the same paradox applies.

“Being informed of” is different than “deduce”.

4. > The essential part here seems to be the “paradox” condition of the prisoner’s reasoning, which works like an extra day.

You’re getting close to Smullyan’s argument, which is that there are _two_ parts of the statement. A: The hanging will happen at such a time, and B: The hangee will be surprised.

5. Solution 1: Keep the prisoner from knowing what day it is. He will not know what day he is to be hanged before the hanging, because he will not know what day it is. He will not know what day the hanging is after the fact because he will be dead. You could tell him the day during the hanging without violating the premise, but it’s not required by the way the problem is worded.

Solution 2: There is a definitional issue. What constitutes ‘before it happens’? If ‘it’ is when his neck snaps, he can be prepared for hanging before midnight, and the trap door released close enough to midnight that he can’t tell on which day he’ll reach the end of the rope. If some other definition of ‘it’ is used, whatever constitutes ‘it’ can be similarly timed.

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