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.)
Best Answer
The specific example you posted is technically a riddle:
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:
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:
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:
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:
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:
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.