# is the missing dollar unanswerable

here is a little riddle that i heard once and told it a thousand times. The thing is I don’t know the answer, and nobody I’ve asked does either.

here it is:

3 guys arrive in NYC–they look for the cheapest place to stay for the night. It’s \$30 and they each pay \$10. They get to the room, and a few minutes later there is a knock on their door. It’s the person that checked them in. He says "Sorry I forgot that on thursdays all rooms are \$25 not \$30. He hands 5 one dollar bills to the person that answered the door, and that person in turn tips the dude \$2, and gives the other two \$1 and \$1 for himself. Now if each paid \$10 and each got a \$1 refund that equals \$9 that each person paid for that 1 room. \$9 times the three of them equals \$27. The attendant got a two dollar tip. \$27 plus \$2 equals \$29. What happened to the other dollar?

Just ask yourself “where’s the money now?”

Each man has \$1, check-in person has \$2, cash register has \$25.

Add those up and you get \$30.

Put another way, each man paid \$9. 3 X 9 = 27. Subtract the \$25 the hotel got for the room. (3 X 9) - 25 = 2. The bellboy has \$2. (3 X 9) - 25 - 2 = 0. Zero dollars missing.

As my dad once said, “Figures don’t lie, but liars figure.”

but thats the whole thing–why doesnt it add up in the riddle. The number 9 is weird, and there are other riddles concerning the number 9, but I cant think of them though. Of course if the room was 25\$ and the check in person got \$2 and each occupant got \$1 back it equals \$30.

The whole point of the riddle is that if you start with \$30 and spend \$30 then you get \$5 back and spend \$2 --whats the total? \$3. Divide the \$30 by 3 and you have \$10 each subtract the 1 and its 9. What happened to the other dollar. Every is accountable in the riddle.

The hotel got \$27
the guys got back \$3
27 + 3 = 30

The boys are out \$9 each x 3 = \$27
The hotel got \$27
27 - 27 = 0

zero sum gain the two dollars is included in the \$27 dollars the hotel got, so these 27 + 2 dosen’t make a valid desription of the transaction.

watch what you say
or they’ll be calling
a liberal,fanatical
a criminal…

to put it in a slightly different way:

\$27 = the final total amount the men paid
\$2 = the part of the \$27 that the bellhop got
\$25 = the part of the \$27 that the hotel got

\$2 + \$25 = \$27
Adding the \$1 that each man has left of his original \$10:

\$27 + \$3 = 30

No missing money.

It doesn’t make sense to add the \$2 to the \$27, because the \$2 is already included in the \$27, i.e., you’re counting the \$2 twice.

This problem is confusing because the meaningless \$29 total just happens to be close to the original \$30. Fiddling with the original amounts can increase the difference between those numbers.

But where were the Spiders?

Cecil has addressed this, as well (of course):

http://www.straightdope.com/classics/a910621.html

The transactions were made with addition/subtraction. The way the riddle is worded they throw in multiplication. Multiplication is performed first in mathmatics, so it misleads the listener into thinking there’s a missing dollar.

The transaction is 3 X 10 - 5 + 3 + 2. Three men pay \$10 (3 X 10). The hotel can only charge \$25 so \$5 is subtracted from the bill (3 X 10 - 5). One dollar goes back to each of the three men (1 X 3). The bellboy gets \$2. So the equation is 3 X 10 - 5 + 1 X 3 + 2 = 30. The riddler wants you to think that it’s 3 X (10 - 1) + 2. But can’t do that because multiplication has to take place before addition or subtraction. Also, it makes you believe that the amount in question is \$30 instead of \$25 by using the \$3 and the \$2. As you can see, 3 X (10 - 1) + 2 = (3 X 9) + 2 = 29. But look at the earlier equation. There’s a 5 in there that isn’t accounted for.

The point is, you can’t subtract 1 from 10 before you multiply 10 by 3. 10 X 3 = 30. Each man got \$1 back. 30 - (3 X 1) = 27. The bellboy kept \$2. 30 - 3 - 2 = 25. \$25 is the cost of the room. 3 X 10 - 5 + 3 + 2 = (3 X 10) - 5 + 3 + 2 = 30 - 5 + 3 + 2 = 30 - 5 + 5 = 30 + 0 = 30.

jaydabee, zgystardst, and tomndebb:

Your explanations were much more elegant than mine.

tomndebb i did a search to make sure i wasn’t asking a question that had already been answered, i couldn’t find one in the serach results—sorry

The search engine is a bit cranky. It took me two tries (but I remember a lo-o-o-o-n-n-ngg thread discussing whether Cece got it right on the AOL boards). It took me a couple of tries this time, too.

I’m probably being too touchy: this has been the fifth issue that appeared in the last three days that Cece has covered. Sometimes it seems as though no one is checking the archives.

Tom~

Johnny, not only were the other explanations more elegnant, yours was pretty much off.
The multiplication aspect has nothing to do with it. The problem lies in subtracting where they should be adding (or, depending on how you’ve set it up, adding where they should be subtracting.)

CKDextHavn,
Maybe. But it all adds up. Whatever works.

The point is that the original “problem” is mathematical gibberish. Compare it to this one:

2 x 3 = 6
2 + 3 = 5
What happened to the missing one?

While somewhat more complicated, the hotel riddle is no more sensible.

I use this puzzle as an example of why people should study algebra. In one sense, people have an intuitive algebraic sense. But unless you’re trained in formal algebra it’s hard to see how this puzzle tricks you into doing the algebra wrong.

I don’t think the multiplication analogy is apt. Rarely do people confuse addition with multiplication, but commonly they confuse addition with subtraction. Ever see the old Abbott and Costello routine about changing a twenty dollar bill? Sure, four fives. Oh, wait, let me give you two fives for a ten…
Etc.

The Abott & Costello routine was hilarious! Do you have the entire text? (IIRC, it took place in an army barracks when Abott wanted money to go out on the town.)

Of course it’s easy to show how the figures really add up, but the OP was asking why the riddle doesn’t work as stated. The simple answer: instead of adding the \$2 tip, you should have subtracted it. They each paid \$9, which adds up to \$27. Subtract the \$2 tip, and you get the \$25 paid to the hotelier.

My uncle pulled this one on me when I was around eight or nine years old, and it bothered me for years. I think I was in high school when I finally figured it out. It’s been around a while - I turned nine in 1970.

For the more advanced nephews/nieces, there’s another problem I like to use. You have twelve coins, one of which is counterfeit. The only way to tell whether it’s counterfeit is to compare its weight to the others - it doesn’t weigh the same, but we don’t know if it’s more or less. You have a balance to do the job, and you get only three weighing events to figure out which coin it is, and whether it weighs more or less.

I was finally able to figure this one out when I looked at it from an information theory point of view. How much information do you need, and how much can you get out of each weighing?

Here’s the missing dollar:

Two women arrive at a hotel and the desk clerk tells them that a room is \$30.00 so each woman gives him \$15.00. Later on, he finds out that the most he can change for the room is \$25.00 so he sends the bellboy up with \$5.00. On the way, the bellboy keeps \$3.00 and gives each woman \$1.00. This means that each woman paid \$14.00 for the room. 2 x \$14.00 = \$28.00 plus the \$3.00 that the bellboy kept makes \$31.00. But there was originally \$30.00. Where did the missing dollar come from?