Unexpected hanging paradox

The unexpected hanging paradox is a paradox involving logic. It is alternatively known as the hangman paradox, the fire drill paradox, or the unexpected exam paradox.

The paradox

A judge makes two statements to a condemned prisoner:

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• You will be hanged at noon one day next week, Monday through Friday.
• The execution will be a surprise to you: you won't know the day of the hanging until the executioner knocks on your cell door at noon that day.

The prisoner reflects on these statements, and then smiles. "If the hanging were on Friday," he thinks, "then it wouldn't be a surprise; for I would know by Thursday night that I was going to be hanged on Friday, since no hanging had yet occurred and only one day was left. So the hanging can't be on Friday."

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"But," he continues, "then the hanging can't be on Thursday either. If it were, then it wouldn't be a surprise either. For I would know on Wednesday night that I was going to be hanged on Thursday, since no hanging had yet occurred and only two days were left, one of which (Friday) I already know cannot be execution day. So a Thursday hanging is impossible too."

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Similar reasoning shows that the hanging can't be on Wednesday, Tuesday, or even Monday! He returns to his cell confident in his safety: "I have been sentenced not just to a hanging, but to a surprising hanging and one that must occur next week; but no hanging meeting both conditions can be carried out."

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The next week, the executioner knocks on his door at noon on Wednesday - an utter surprise. Everything the judge said has come true. Where is the flaw in the prisoner's reasoning?

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A simpler form of the paradox

To gain some insight into this problem, it's helpful to look at a simpler form of the paradox, which reduces the number of days to one (Friday). In this version, the judge's sentence is:

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• You will be hanged at noon next week on Friday.
• The execution will be a surprise to you.

The prisoner exclaims, "No hanging can meet both conditions of the sentence. How can it be a surprise, if you've already told me it will be on Friday? That's contradictory! Therefore, the sentence as you have made it, your honor, cannot be carried out. The executioner cannot hang me in a way that is consistent with your sentence."

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The next Friday, the prisoner is hanged. This comes as a surprise to him, since he had convinced himself that the sentence could not be carried out. What was wrong with his reasoning?

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