Three men go to stay at a motel, and the man at the desk charges
them $30.00 for a room. They split the cost ten dollars each. Later the manager
tells the desk man that he overcharged the men, that the actual cost should
have been $25.00. The manager gives the bellboy $5.00 and tells him to give
it to the men.
The bellboy, however, decides to cheat the men and pockets $2.00, giving each
of the men only one dollar.
Now each man has paid $9.00 to stay in the room and 3 x $9.00 = $27.00. The
bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00  so where is the missing $1.00?
This question has been sent to Dr. Math many times. Here's a sampler of
answers from a variety of 'math doctors':
 From Doctor Ethan:
The problem is that the question is always cleverly phrased to conceal what
is really going on. Since I don't want to just give you the answer, I'll tell
you how I think about it and then you can see if you understand it.
First let's locate all that money. There are two ways to think about how much
money is there, and the trick in this question is that it combines the two ways:
 How much money did the men originally pay?
 How much money did they end up paying?
For (a), we need to account for $30. The owner keeps $25, the bellboy gets $2,
and the men get $3 back. That adds up fine.
Now let's look at (b). How much money did the men end up paying? $27, of which
$25 went to the owner and $2 to the bellboy. That adds up too.
The problem with the question is that the $2 the bellboy gets is already contained
in the $27 that the men end up paying, so we shouldn't expect adding that $2 to
anything to be meaningful.
 Dr. Rob says:
Since each man has now paid $9 for the room (3 x 9 = 27), and the bellboy has
$2 in his pocket (27  2 = 25), the rest of the money is in the hotel till.
The trick is to realize that the $2 has to be subtracted from the $27, not
added to it.
 Dr. Wilkinson adds:
"...three nines are $27, plus the $2 which the bellboy got is $29. Where did
the extra dollar go?"
Be careful about accepting what you are told! The flaw is in the phrase
"plus the $2 which the bellboy got." This should not be added; it should be
subtracted, since the $2 the bellboy got is part of the $27 dollars the three
men spent altogether. If you subtract the $2 from the $27 you get the $25
that goes into the till.
 Dr. Pete elaborates:
Write out a table: Deskman Bellboy Men

$0 $0 $30 < men have not yet paid for room
$30 $0 $0 < men pay deskman
$25 $5 $0 < deskman pays bellboy
$25 $2 $3 < bellboy stiffs men

$25 $2 $27 < what each group of people has
after all the transactions
Here, the last row is simply the difference between row 4 and row 1. In all
but the last row, the sum of the dollar values along each row is constant and
equal to $30. In the last row, the apparent fallacy is that the men and the
bellboy should have 30 dollars between them, but this statement is false, as
it obviously ignores the question of what the deskman has. In fact, the correct
statement about the last row is that the sum of what the deskman and the
bellboy have must equal the debt of the three men.
The men have collectively paid 27 dollars for the room, which is obvious,
since the bellboy took $2 and the actual cost was $25. And so we see that
there is no missing dollar, because the $27 the men paid is a debt, written
as a negative number, and the $2 the bellboy took is a profit, which
is a positive number, and the sum is not $29, but a debt of $25, which
was paid to the deskman.
To exaggerate the example, suppose the cost of the room is $5, the bellboy
taking $22, the men getting $3. Then it becomes clear that the $27 that the
men wind up paying for the room "plus" the $22 the bellboy takes just doesn't
equal anything meaningful. What's going on is that $22 of the $27 that the men
pay has wound up in the bellboy's pocket, so adding $22 to $27 is in essence
counting the bellboy's money twice.
 Dr. Rothman numbers the dollars:
Let's give each of the $30 a number from 130, keep track of each individual
dollar, and see how the problem works.
Dollars numbered 130 are given to the manager. Then he wants to give $5 back,
so he keeps the dollars numbered 125, and gives numbers 2630 to the bellboy
in the form of a five dollar bill. The bellboy splits up the five to get 5 one's:
numbers 26, 27, 28, 29 and 30. He gives numbers 26, 27 and 28 to the customers
and keeps numbers 29 and 30 for himself.
