The Unexpected Hanging Paradox

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The Unexpected Hanging Paradox

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 Wednesday, surprising the prisoner.

Khan’s Solution: There were a total of five time-lines between Monday and Friday and each day, Monday, Tuesday, Wednesday, Thursday and Friday was the last day in each timeline, (1), (2), (3), (4) and (5) accordingly. See the Fig.1 below;

The five time-lines from the Fig.1

(1) Monday.

(2) Between Monday and Tuesday.

(3) Between Monday and Wednesday.

(4) Between Monday and Thursday.

(5) Between Monday and Friday.

According to the Judge’s conditions, the prisoner was simply imagining the last day (one by one) of each of the above five time-lines between Monday and Friday, that if the Judge could pick (starting with the timeline, 5) would not enable him to be hanged unexpectedly on the last day. The prisoner failed to realize that it was not a question of if the Judge could pick, rather it was a question of which one of the five time-lines would the Judge finally pick. For example, the Judge could simply ask the prisoner, “Which timeline will I pick for you between Monday and Friday?”, and the prisoner would have no way of knowing.

Thus, the Judge simply decided to hang the prisoner on Wednesday to his utter surprise. This was the last day in the timeline, (3) between Monday and Wednesday. That day, according to the prisoner’s logic, he was supposed to know of the day earlier. Besides, by picking the timeline, (3), the Judge wiped out Thursday and Friday when the prisoner’s imaginary logic was supposed to begin.

Conclusion: the actual meaning of it all was that the prisoner’s surprise finally depended on the Judge’s unpredictable choice of the timeline that the prisoner failed to take into count. Copyright © 2011 Nasser Khan

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