The unexpected hanging is a remarkable logical paradox that appears to have begun circulating by word of mouth in the 1940s, often in the form of a puzzle about a man condemned to be hanged.

 

A judge, with a reputation for reliability, tells a prisoner on Saturday that he will be hung on one of the next seven days but that he will not know which day until he is informed on the morning of the execution. Back in his cell, the prisoner reasons that the judge must be wrong. The hanging cannot be left until Saturday, because the prisoner would certainly know, if this day dawned, that it was his last. But if Saturday is eliminated, the hanging cannot take place on Friday either, because if the prisoner survived Thursday he would know that the hanging was scheduled for the next day. By the same argument, Thursday can be crossed off, then Wednesday, and so forth, all the way back to Sunday. But with every other day ruled out for a possible unexpected hanging, the hangman cannot arrive on Sunday without the prisoner knowing in advance. Thus, the condemned man reasons, the sentence can't be carried out as the judged decreed. But then Wednesday morning comes around and, with it, the hangman – unexpectedly! The judge was right after all and something was awry with the prisoner's seemingly impeccable logic. But what?

 

More than half a century of attack by numerous logicians and mathematicians has failed to produce a resolution that is universally accepted. The paradox seems to stem from the fact that whereas the judge knows beyond doubt that his words are true (the hanging will occur on a day unknown in advance to the prisoner), the prisoner does not have this same degree of certainty. Even if the prisoner is alive on Saturday morning, can he be certain that the hangman will arrive?

 

References

1. Chow, T. Y. "The Surprise Examination or Unexpected Hanging Paradox." Amer. Math. Monthly, 105: 41–51 (1998).
2. Quine, W. V. O. "On a So-Called Paradox." Mind, 62: 65–67 (1953).