Unexpected Hanging Problem

Judge Wright has a reputation for always being correct. Standing before him is a condemned prisoner, which turns out to be a logician. Judge Wright decides to have some fun with him, and says, “You will be hanged at noon on one day of the coming week, and it will come as a surprise to you. You will not know until the executioner comes knocking on your cell door at 11:55am the day of the execution. It is 4pm Monday, so it will happen by noon next Monday at the latest.”

The prisoner carefully considers Judge Wright’s comments. He reasons that, next Monday, the last of 7 days, cannot be the day of execution, because being the last possible day of execution, what kind of surprise would that be? That rules next Monday out completely. How about Sunday? Well, Monday is out, so Sunday is now the last possible day of execution. But again, it wouldn’t come as a surprise either. So Sunday is also out. By similar reasoning, Saturday, Friday, Thursday, Wednesday, Tuesday are all logically ruled out. The only conclusion the prisoner could come to, is that Judge Wright made a rare mistake, and he will not be hanged at all.

At 11:55am Thursday, the executioner came knocking on the cell door. And sure enough, it came as a total surprise to the prisoner. Judge Wright had been right all along.

This is a rare example of a paradox that is also humorous. Although it does not initially seem worthy of serious discussion, surprisingly enough no fewer than 200 papers have been published on this paradox. Naturally, many of them start by dismissing other views and claiming that theirs is the long-awaited solution, the final nail in the coffin.

What exactly, is wrong with the prisoner’s reasoning? There are two main approaches to resolve the paradox, logical and epistemological. The logical approach breaks down the argument by examining the basis (axiom) used for the reasoning to:

The prisoner will be hanged next week and its date will not be deductible in advance by using this announcement as an axiom.

Which is a self-referential statement and cannot be used to construct a valid argument.

The epistemological argument focuses on the meaning of the announcement, specifically the “surprise” part. Rather than explaining it, I will use this brilliant variant of the paradox (by R.A. Sorensen):

Exactly one of five students, Art, Bob, Carl, Don, and Eric, is to be given an exam. The teacher lines them up alphabetically so that each student can see the backs of the students ahead of him in alphabetical order but not the students after him. The students are shown four silver stars and one gold star. Then one star is secretly put on the back of each student. The teacher announces that the gold star is on the back of the student who must take the exam, and that that student will be surprised in the sense that he will not know he has been designated until they break formation. The students argue that this is impossible; Eric cannot be designated because if he were he would see four silver stars and would know that he was designated. The rest of the argument proceeds in the familiar way.

I could not possibly come up with a better example. Not only does it highlight the subtle, different meanings of “surprise”, but more importantly the absurdity of the chained argument when “surprise” is defined properly. An elucidating example requires a deep understanding and effective communication. It cannot be faked.

Note: for a more detailed explanation and a comprehensive list of articles, I strongly suggest Timothy Chow’s paper