The Unexpected Hanging Paradox
Stimulating critical thinking - a loose series on paradoxes
Gödel's is a newsletter about interweaving ideas and making decisions under uncertain conditions. I discuss knowledge management, mental models, and supporting Tools for Thought.
Imagine a prisoner is sentenced to death. The judge, known for his eccentric ways, tells the prisoner, "You will be hanged next week, but the exact day of your hanging will be a surprise." The prisoner, taken back to his cell, starts pondering the judge's words.
He reasons that he cannot be hanged on Friday, the last day of the week, because if he were still alive by Thursday night, there would be no surprise – Friday would be the only day left. Therefore, he concludes that the hanging cannot happen on Friday.
Continuing this line of thought, the prisoner then considers Thursday. Since he has already established that Friday is not a possible day, Thursday would not be a surprise if he were still alive by Wednesday night. So, he rules out Thursday as well.
Applying this reasoning further, he excludes Wednesday, Tuesday, and Monday. Satisfied, the prisoner concludes that the hanging cannot happen at all since it can't be a surprise on any day of the week.
However, the paradox unfolds when the executioner unexpectedly arrives on Wednesday, catching the prisoner by surprise. The paradox lies in the fact that the prisoner logically deduced he wouldn't be hanged on any day, yet his execution still occurs on a day that surprises him.
This paradox presents a fascinating puzzle. It hinges on the concept of what constitutes a 'surprise.' The prisoner's reasoning seems sound – if he can predict the day of his execution, it wouldn't be a surprise. Yet, his conclusion that he cannot be hanged on any day is wrong, as evidenced by his surprise at the executioner's arrival.
Source: “Mathematical Games” in Scientific American Magazine Vol. 208 No. 3 (March 1963), p. 144