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A friend of mine posted a riddle on Facebook involving adding money and then subtracting money. It ended with a question asking where $1 went, but the trick was that there was no discrepancy, so the $1 didn't exist.

The question: Three guys walk into a hotel, and they're going to split the cost of a room. The room is $30. They each kick in $10 and head up to their room. The manager gets wind of it and tells the clerk the room is only $25. He hands five $1 bills to the bell hop and tells him to go refund the guys' money. On the way up to the room, the bell hop gets to thinking, as bell hops are wont to do, and says to himself, "No way can three guys split $5, I'm going to help out." He stuffs $2 in his pocket, knocks on the door, gives each guy back a buck and heads back downstairs to the desk, glowing in the warmth of a job well done. So now each guy has paid $9. $9 times 3 is $27 plus $2 the bell hop stole--only $29! Where is the other buck?

(I bolded the question)

I said the question was invalid, but my friend pointed out that every question is valid to ask.

What would I call this question that has no answer? (If I understand correctly rhetorical is when you can answer the question, but it is not meant to be answered, so by that definition "rhetorical" would be invalid here.)

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Must resist the temptation to flag this as "not a real question"... :) –  phenry May 27 '11 at 15:10
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Just to point out the fallacy for anyone who hasn't encountered this already (and can't figure it out): the $2 the bellboy pocketed is the difference between what they ought to have paid, $25, and what they actually paid, $27. Adding the $2 to the $27 is a nonsensical operation. –  Marthaª May 27 '11 at 15:22
    
@Martha I was confused, thanks for that :) –  Xan May 27 '11 at 18:24
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I don't know what to call the question but I tend to agree with MrHen; I would probably refer to it as a fallacious question (though that is not one word). I believe the correct answer to your friends question though is "Mu" (to indicate a question whose "answer" is to un-ask the question, indicate the question is fundamentally flawed -Wikipedia). As to the assertion that all questions are valid, you might ask your friend why he left his shoes untied (assuming of course that he did not). Or the more traditional question, "when did you stop beating your wife?". –  logicbird May 27 '11 at 20:01
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An easy way to point out the fallacy created by the question is to say "what if the bellhop 'helped out' by just keeping the whole $5? Now the guys paid $30, plus the $5 the bellhop stole makes $35. Where'd the extra $5 come from?" –  Hellion May 28 '11 at 3:38
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12 Answers

up vote 33 down vote accepted

The specific example you posted is technically a riddle:

A riddle is a statement or question or phrase having a double or veiled meaning, put forth as a puzzle to be solved.

That doesn't really answer the more generic question in your title, however. Nor does it address your friend's assertion that all questions are valid. Strictly speaking, not all questions are valid:

What grik plah mot?

This is completely nonsensical regardless of it being phrased as a question. Even restricting oneself to valid English words it is possible to construct invalid questions:

What does the color 9 smell like?

These forms of questions can be described as fallacious. There is an entire list of appropriate fallacies but the most common in terms of questions would be:

  • false dilemma — presenting two options as if they are the only choices available
  • loaded question — asking a question such that any answer would be false or misleading
  • false assumption — beginning your question with an assumption or assertion that is untrue
  • non sequitur — asking a question such that the question makes no sense given the description of the scenario

The $1 riddle is an example of a non sequitur. The logic within the puzzle does not flow properly and the question at the end does not follow from the given scenario. Here is Wikipedia's description of the fallacy:

The term is often used in everyday speech and reasoning to describe a statement in which premise and conclusion are totally unrelated but which is used as if they were.

Their article more directly addresses the form related to formal logic but the principle holds. Here is the (slightly trimmed) riddle with the non sequitur highlighted:

Three guys split the cost of $30. They each kick in $10. The manager tells the clerk the room is only $25. He hands five $1 bills to the clerk and tells him to refund the money. The bell hop says to himself, "No way can three guys split $5, I'm going to help out." He stuffs $2 in his pocket, knocks on the door, gives each guy back a buck. So now each guy has paid $9. $9 times 3 is $27 plus $2 the bell hop stole--only $29! Where is the other buck?

Asking where the other buck is has nothing to do with the exchanges that happened earlier in the question.


By the way, as Martha pointed out in the comments, the correct action would have been to subtract the bellhop's $2 which arrives at $25, which is the cost of the room.

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11  
Colorless green ideas sleep furiously. –  Charles May 27 '11 at 16:20
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+1 this answer smells terrific! –  BlueRaja - Danny Pflughoeft May 27 '11 at 18:25
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I agree. Thanks a lot for this answer. It is perfect. I'm espescially glad you included some other fallacies that at the very least seem like they represent the question. –  Jason McCarrell May 27 '11 at 18:54
    
+1 for Chris Rice. –  John May 28 '11 at 6:08
    
@MrHen, very nice. The question itself could also be called a loaded question (yes, it is a part of non sequitur, but when the question implies a problematic assumption I feel this is a more specific term). @Jason McCarrell - one note, possibly too obvious, such questions are fallacies in a sense of a part of a rhetorical speech that tries to rightfully persuade the audience of an issue. In the context of a riddle asking such a question is not a fallacy, but a part of the riddle. –  Unreason May 30 '11 at 8:01
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In this case it's a trick question, because it intentionally misleads. If a question is invalid for some other reason, like being honestly based on what turned out to a false premise, then I think invalid is correct.

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Even though I support your answer let me nitpick: the question is read as an 'honest question' until you realize what the solution to the problem is. Using your terminology: when I was solving the problem it was an honest question for me at first; then I realized that it was based on a false premise - this would qualify it as invalid question, but at the same time I also realized that it is a trick question. –  Unreason May 30 '11 at 8:20
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@Unreason: No, it was always a trick question, you just didn't realise until you solved it. That is the point of a trick question. –  Marcin May 30 '11 at 12:51
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@Unreason: from the perspective of the asker it's a trick question (or, in other cases, invalid or based on false premises). The person being asked the question doesn't have all the information to make a judgment initially. –  Monica Cellio May 30 '11 at 13:20
    
@Marcin, @Monica Cellio, I mark your comments as good, as they are valid; however I did not overlook them - you both are speaking from a position of absolute objectivity, which is not enough to consider when looking at the linguistic aspect of semantics (esp. Monica, when you imply subjectivity - "...being honestly based on what turned out..." which implies imperfect knowledge and exclude it with "intentionally" - which implies perfect knowledge. Exclusion is through: "If ... is invalid for some other reason..."). –  Unreason May 30 '11 at 14:55
    
@Unreason: That has nothing to do with it. The term "trick question" is as a matter of English lexicon applied to a trick question whether or not the person being asked knows it is a trick question. This word does not refer to the listener's state of knowledge. Your argument could be applied to literally any word - you might as well say that you should be able to refer to a piece of cheese you had mistaken for a rock as a rock because in your mind it was a rock. –  Marcin May 30 '11 at 15:10
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This is exactly the case of sophistry. Because he knows that he is playing a trick.

Sophistry: the clever use of reasons or explanations that seem correct but are really false, in order to deceive people

If he didn't know that he is playing some trick on you I would call it fallacy.

Fallacy: a weakness in someone’s argument or ideas which is caused by a mistake in their thinking

If it was a known problem that was accurately stated, and has not yet been solved, I would call it Open Problem.

Open problem: In science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved (no solution for it is known).

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I don't think sophistry describes the example question at all. –  Marthaª May 27 '11 at 15:27
    
@Martha I think it might answer this question though, if either one of you is interested. –  KitFox May 27 '11 at 15:33
    
@Martha: Why do think so? –  Manoochehr May 27 '11 at 15:34
    
@Martha: I think a trick question can be one form of sophistry. –  Mitch May 27 '11 at 16:00
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@MrHen: precisely. Sophistry applies to an argument, not to something that is, at its heart, just a joke. –  Marthaª May 27 '11 at 18:35
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It is a loaded question because it presupposes a false and misleading premise. Here the amounts paid plus one amount received are added, when since they come from "opposite sides of the ledger" they should if anything be subtracted. (Amount $27 paid by three lodgers minus $2 pocketed by bellhop minus $25 retained by hotel reconciles to zero.)

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I don't know if there's a specific word for that. Broadly, it would seem to fall into the category of fallacy ("an often plausible argument using false or invalid inference").

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I deleted my previous comments, because I now agree with your friend.

The series of events your friend described did involve fallacious reasoning. The reasoning was invalid. However, the actual question was "Where did the $2 go?", which has a legitimate answer, "The $2 never existed. It was created through erroneous reasoning in the story."

A legitimate, valid question, and a legitimate, valid answer.

So your actual question is really: "How do you describe reasoning which is invalid and leads you to a false conclusion?", for which there are many answers.

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You might consider this question a conundrum, Thefreedictionary calls "A paradoxical, insoluble, or difficult problem; a dilemma."

There is also a sense in which the question is a false dilemma: the implied solution is that the dollar is either "missing" or "not missing;" both "missing" and "not missing" are removed from consideration; the misleading suggestion leads me to a third possibility, following.

You might consider this kind of question a red herring. The question "Where is the other buck?" is a deliberately misleading question.

I might gently disagree with your friend's assessment that "every question is valid to ask." Cross-examining a witness with "Do you still beat your wife? Answer 'Yes' or 'No'!" is not a valid question when the witness has never beaten his wife.

The question "Will the next word you say be 'No'?" can only be answered with "No," which would be a falsehood, or something other than "No," which would also be a falsehood. A question that cannot be answered is not a valid question.

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I would call it misleading because it is not valid. The question is not valid because the premise is not valid. Claiming something is valid does not make it valid. The friend made a statement without any evidence to back it up.

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Were you thinking of paradox ?

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A paradox is something else entirely. –  MrHen May 27 '11 at 16:24
    
The example is presented as a paradox, I don't think it's the best answer - but it might have been what the OP was searching for, or at least it might be useful to someone. –  mgb May 27 '11 at 16:36
    
Ehh... well, okay. The vote is locked in already, though so I cannot revert it. It wouldn't be bad to include a definition or link, though, so I'll take the vote back if someone feels like editing. –  MrHen May 27 '11 at 16:39
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I believe that the question can be categorized as brain-teasers. When the question is not valid at all and it baffles understanding and cannot be explained.

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  • illogical
  • nonsense
  • absurd
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It could be

  • enigma
  • trick question
  • unsolvable question
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Could it be "unsolvable" when the question leads misunderstanding ? –  Gigili May 27 '11 at 17:02
    
@Giglili, in a sense of tricky question, maybe not, but in the sense of question with no answer: yes. Just like enigma is in the sense of something with no known answer (even though even solved enigma seem to keep its name) –  Unreason May 27 '11 at 18:18
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protected by RegDwigнt May 27 '11 at 16:49

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