# The Unexpected Hanging Paradox

30.08.2018

Episode #10 of the course Brain-twisting paradoxes by John Robin

Welcome to our final paradox. As promised, the stakes are high today!

Let’s start with a story.

A prisoner, a judge, and an executioner meet at dawn on a Sunday.

The judge, having heard enough evidence, passes a sentence: The prisoner is guilty and will be hanged. There is no date set. Rather, the judge is cryptic in how he speaks:

“At a dawn, sometime in the next five days, you will be surprised as you are awakened, dragged from your cell, then brought to the gallows, where you will be hanged.”

The executioner leaves the trial eagerly, while the prisoner is taken back to his cell in dread. Yet for him, there’s still some hope. There might just be a way out of this, if he can channel his skills. After all, he knows lots about brain-twisting paradoxes and senses his situation might be one of them!

Do you spot what it might be? Take a moment to think about it. Imagine it’s you instead of the prisoner. How would you escape?

A Busy Sunday Afternoon

Our prisoner spends the remainder of Sunday wracking his brain.

First of all, he is able to determine that his day of possible death will be either Monday (within the next day), Tuesday (within the next two days), Wednesday (within the next three days, or Thursday (within the next four days). It won’t be Friday because that’s the last day, and if he isn’t hanged by Thursday, he’ll know for certain he’s being hanged on Friday, which means he won’t be surprised as the judge said he’d be.

Now, he works out that it can’t be Thursday either, since by logic, the judge will know if the prisoner doesn’t expect to be hanged on Friday, then he will expect that after Wednesday’s dawn passes, the only day left is Thursday.

This leaves the prisoner only with Monday, Tuesday, or Wednesday morning as possibilities. But he is able to reason by the same logic that Wednesday is out for the same reason Thursday was: If he expects Monday through Wednesday are his only possibilities, then by late Tuesday, he’ll be expecting to be hanged Wednesday morning; therefore, the day has to be Monday or Tuesday. The exact same reasoning applies for Tuesday.

This leaves him finally expecting to be hanged Monday—ah, but now there’s a new problem: He’s expecting to be hanged by Monday, so it can’t be Monday either.

This leaves him with one conclusion: The judge’s cryptic sentence is another way of saying he will not hang him at all, since all five possible dawns are ones he will expect. Phew! Now he can relax and enjoy the rest of his Sunday and expect Friday he’ll be let go.

Surprise!

Wednesday morning comes, then—surprise! The guards awaken him, drag him away, and he’s hanged.

He wasn’t expecting that at all!

Aside from seeing that understanding brain-twisting paradoxes isn’t always a get-out-of-jail-free card, you can appreciate how this paradox shows the self-contradictory nature of statements.

This paradox has been debated by philosophers (a part of the broader Newcomb’s paradox). The key issue here is the interpretation of the word “surprise.” Were our prisoner to have been a bit more diligent, he would have realized that he was falling into his own trap by thinking the judge would not hang him at all, since that logical conclusion is what the judge would have been expecting him to arrive at, thus making the element of surprise all the more surprising. The only way out for him would have been to realize that indeed, he’s trapped in a brain-twisting paradox and he better get out—time to call in those favors and engineer a late Sunday jailbreak!

Words can trap us in their own constructions, as this paradox and many like it demonstrate. This has far-reaching ramifications in anything that depends on interpretation, law and contracts being one of them. Your take-home from this final lesson is to be careful when you think you have understood all possible implications of a statement, because there’s always room for surprises.

I hope you’ve enjoyed this course and have a newfound appreciation for the paradoxes that lurk in the many aspects of our life. We have just scratched the surface, and now that you’re curious, I encourage you to read further and venture into the brilliant world where the mind and reality twist in unexpected ways.

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Recommended reading

Read more about this paradox in the article, “Mathematical Games,” by Martin Gardner.

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