TaBoo's Punishment Paradox

The First Riddle

Published on February 7, 2005 By TaBoo Tenente In Entertainment

Note: I have heard this riddle presented in many forms. Most recently, I read this version in LABYRINTHS OF REASON by William Poundstone.

The jury found the prisoner guilty of the worst sort of murder. Well within his rights to sentence the prisoner to death, the judge could not indulge himself and have the manner of the murderer's death exceed a simple hanging. Nevertheless, the nature of this specific crime filled the judge with outrage, and he struggled within the bounds of his mandate to find a way to make this particular hanging slightly more painful.

The day of the sentencing arrived, and the prisoner, along with his counselor, appeared before the judge.

The judge said, "I am tempted to order your immediate execution and wash my hands of this heinous crime once and for all. However, I find you to be a heartless, cold human being, and nothing would please me more than to pierce your hardened caul. Therefore, here is your sentence: Because an immediate hanging would cut your anxiety short, while an extended wait would only allow you to enjoy your life longer than you deserve, I sentence you to hang at sunrise on one of the seven days next week. I further instruct your executioner to make sure you have no way of knowing in advance what day you shall die. Thus, you will approach each night's sleep with the fear that this may be your last night on earth. Then, come sunrise, the executioner will walk you to the gallows, hang the rope about your neck. Perhaps he will then remove the noose and return you to your cell for another day. Or perhaps the door below your feet will vanish, and you will die. However it should happen, it will come as a complete surprise, I guarantee you this, upon my honor and my life. You will not know the moment of your death."

The murderer found himself startled and suddenly afraid of the cruelty of the sentence. But to his anger, he found his counselor smiling with relief. The murderer turned then and snarled, "What happiness can you find here?"

"Well, it's just this: the judge cannot hang you now, or ever!" the lawyer crowed. "Consider: you are supposed to hang one sunrise in the next seven, but you cannot know for certain which day. So they cannot hang you next Saturday morning. It is the last day within the sentencing period, and if you walked to the gallows on that day, you would know you would hang. That very knowledge was prohibited by the judge himself. Are we agreed? You cannot hang on Saturday."

The murder nodded his head, and said, "So? Who cares? It will not be Saturday, but it might be Friday or Monday or any other day."

The lawyer shook his head. "Suppose you live to see Thursday afternoon. You will realize then that the only remaining mornings are Friday and Saturday. However, you already know it cannot be Saturday. But if it must be Friday, then it cannot be Friday, either! And now you must see that this holds true for any day of the seven. On Wednesday afternoon you will know that Thursday is the only possible day; and then, of course, it cannot possibly be Thursday! So on, so forth. The judge must release you or break his solemn oath as a judge!"

Slowly the prisoner began to smile. That night he slept like a baby, and walked confidently to the gallows on Sunday morning, and just as he suspected, he walked back to his cell. He walked again on Monday morning, felt the rope secured about his neck, but he smiled nonetheless. Sure enough, the rope was removed, and he strolled back to his cell yet again.

On Tuesday morning the executioner led him to the gallows and secured the noose. That morning gave birth to a piercing sunrise, and silhouetted the murderer like a frame around a photograph, as he hanged by a stout rope--quite unexpectedly.

Did the judge fulfill his sentence?

Copyright ©2004, ©2005, ©2006 Joshua Suchman. All rights reserved.
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Comments

Dr Guy

on Feb 07, 2005

It would appear so, as he was surprised after being told by his lawyer he would never die.

Danny Bassette

on Feb 07, 2005

I think the real question is: was the judge counting on the lawyer to attempt to ruin things?

TaBoo Tenente

on Feb 07, 2005

you both have picked up crucial elements to the paradox. the judge, in order to fulfill his oath, must be SURE. he can't guess or hope that the lawyer would explain the paradox to the prisoner. would the judge know about the conversation between the murderer and the lawyer? and even if the judge did know, wouldn't he still be unable to execute him on saturday, and therefore friday, and therefore thursday. . .?

Dr Guy

on Feb 07, 2005

I think the real question is: was the judge counting on the lawyer to attempt to ruin things?

If that is the case, then it was one very smart judge! Lawyers are not so easily duped, but it is possible.

Dr Guy

on Feb 07, 2005

This gets back to the theory that motion is impossible. Since before you can run 100 yards, you must run 50, and before 50, 25, etc. Such that in order to move at all, you must pass an infinite number of points in what amounts to no time.

TaBoo Tenente

on Feb 07, 2005

that is an interesting paradox, and dissects the notion of time, i think. i never really understood that one. maybe it has to do with splitting infinity? i seem to remember a debate about it in some philosphy course, and im fairly sure i came out on the bunghole side of that one.

this paradox is a knowledge paradox. im not sure that qualifies as a hint, but so it goes.

tbt

Dr Guy

on Feb 07, 2005

this paradox is a knowledge paradox. im not sure that qualifies as a hint, but so it goes.

What came first, the chicken or the egg?

TaBoo Tenente

on Feb 07, 2005

Look at it this way: suppose, instead of allowing seven days during which the prisoner might hang, the judge allows for only saturday or sunday. let us say it is friday night, and according to the judge, the prisoner will either hang on saturday or sunday--and the prisoner is still not to know which day.

does this change the problem, or is it still the same question?

Toblerone

on Feb 08, 2005

He was hanged in the night. The judge forefilled his oath by lying about doing it at sunrise. He said you will not know the MOMENT of your death not DAY.

TaBoo Tenente

on Feb 08, 2005

it's a good try, but the judge does say the prisoner will hang "at sunrise on one of the seven days next week".

i'll offer two possible solutions. the first solution depends on the validity of knowledge. suppose you say to a friend, "i have a hunch that the time is 2:37pm" and then you look at the clock to find you were absolutely correct. did you "know" the time? or suppose i ask you "what is the square root of three rounded to the nearest thousandth?" and, really having no clue but you had just been reading a book about george washington and found that he was born in the year 1732, you say "1.732". you'd be right, but did you "know" the answer?

the first solution hinges on the belief that hunches are not knowledge. in this scenario, the lawyer was incorrect to deduce that if the prisoner were to hang on the last possible day, it still would not be knowledge.

the second solution defines the paradox. suppose the prisoner has no logical skills whatsoever, and the lawyer simply wants to leave the courtroom. the prisoner likely will think to himself "well, my guess is that the judge will hang me at some point early in the week but not right away. " the judge could then hang him any day, and will not contradict the need for surprise. but in the puzzle's scenario, the entire gamut has been laid bare by the lawyer. the prisoner knows he cannot hang on any day, so he will be surprised if he does hang.

the key is this: in this scenario, the lawyer is also wrong to deduce that the prisoner cannot hang on the last morning. knowing that he cannot hang on that day, he can hang.

Dr Guy

on Feb 08, 2005

the key is this: in this scenario, the lawyer is also wrong to deduce that the prisoner cannot hang on the last morning. knowing that he cannot hang on that day, he can hang.

Very good Puzzle! IN the end, the judge had them checkmated. Ignorant, and he could be hung any day. Knowledgeable, the same things since he knew he would not hang.

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