is the missing dollar unanswerable

[siva]

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?

[Arnold Winkelried]

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.

[Johnny L.A.]

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.

[Johnny L.A.]

As my dad once said, "Figures don't lie, but liars figure."

[siva]

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.

[jaydabee]

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
you a radical,
a liberal,fanatical
a criminal.....

[rowrrbazzle]

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?

[tomndebb]

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

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

[Johnny L.A.]

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.

[Johnny L.A.]

jaydabee, zgystardst, and tomndebb:

Your explanations were much more elegant than mine.

[siva]

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

[tomndebb]

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~

[C K Dexter Haven]

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.)

[Johnny L.A.]

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

[Little Nemo]

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.

[Johnny Angel]

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.

[C K Dexter Haven]

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.

[Johnny L.A.]

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.)

[CurtC]

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?

[Henry]

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?

[C K Dexter Haven]

The confusion comes in part from having three characters (hotel, men/women, and bellhop). If there were only two characters, the mystery would vanish immediately:
Three men check in to a hotel, they each pay $10. The hotel clerk realizes the room should only cost $25, but doesn't want to distribute the uneven amount, so gives $1 back to each person and the hotel keeps the extra $2. No problem: hotel has $25 + 2, men have paid $27. No one tries to convince you that the men paid $27 and the hotel kept $2, totalling only $29.

However, this mathematical analysis fails to take into account tax. In fact, the extra dollar goes to the government. What's really missing is that that bellboy should also have paid social security tax on the $2 he kept, about $.14 ... So where's the missing $.14??

[Kat]

In the version I read, the total comes to $63.50, and you have to account for the extra $33.50. But, come to think of it, it doesn't mention Social Security tax...

Click here and scroll down (http://fathom.org/teemingmillions/pofunnies.adp)

[TBone2]

PEOPLE! The "three guys in a rented room" story is a thing of beauty! It has spanned generations and confounded millions! But the secret of it -- the KEY -- is in its telling! NOT in the math or the algebra or in trigonometry or any other studious pursuit. IT'S IN THE TELLING!

One of the hundreds of reasons I have for loving my dad is the fact that he pestered the shit out of me with that riddle for years....


------------------
I don't know why fortune smiles on some and lets the rest go free...

T

[Little Nemo]

2 x $14.00 = $28.00 plus the $3.00 that the bellboy kept makes $31.00.

This is where you went astray, Henry. The three dollars that the bellboy kept came from the twenty eight dollars the two women paid. So why add it back in?

The correct equation is "2 x $14.00 = $28.00 plus the $2.00 that the bellboy returned to the women makes $30.00."

[siva]

I always thought it didn't work because of the fraction involved. After the three guys paid $10 the three of them got $5 back total. The refund is 1.66 each. So subtract the 1.66667 from the original $10 that each guy paid for the room and you have $8.33. If the bellhop got tipped two bucks that means each person pays the bellhop $.66 which in turn leaves each person $7.67. 7.67 multiply that by the three guys and you get $23.01. Add the $2 and you get $ 25.01 that they pay for the room, thats pretty darn close to the $25. The riddle is decieving cuz it doesnt account for the fraction---right?

[siva]

Ohh geez i forgot to say this. after the $5 rebate that left each person 8.33. The 33 cents is whats missing. $.33 x 3 =$1

[C K Dexter Haven]

OK, so the three guys paid $30.
The hotel kept $25, that's $55.
The bellboy kept $2, that's $57.
The bellboy gave $3 back to the guys, that's $60.
The three guys each got back $3, that's $63.
So the three guys really paid $9 each, that's $27 more, added to the $63 gives $90.

Where did the other $60 come from?

The whole question is easy if you say the hotel is in Sweden, since then the taxes account for the extra money, very quickly.

Sheeeeeeeeesh.

[RM Mentock]

Dex

That's gotta go in the rec.puzzles archive! Permission to reproduce? With attribution where possible.

------------------
rocks

[Little Nemo]

Siva, let's make one more attempt here. Get thirty dollars and five envelopes. If you don't have thirty dollars, use thirty pennies or buttons or matchsticks and pretend. Label the envelopes customer 1, customer 2, customer 3, bellboy, and hotel. Draw faces on them if you wish to personalize the whole thing.

Now go through the problem step by step and move the dollars from envelope to envelope as you do so. Hopefully by the time you are done you will have acheived enlightenment and will now understand that there was never more or less than thirty dollars at any point.

[andyrose]

A friend told me that it had to do with limits and trigonometry.

[tomndebb]

it had to do with limits and trigonometry.I presume that this was because the three customers had to cosine for the room? (Or am I going off on a tangent?)

[Wendell Wagner]

No, you just have to look at the problem from the right angle.

[andyrose]

Originally posted by tomndebb
I presume that this was because the three customers had to cosine for the room? (Or am I going off on a tangent?)

Very nicely done.

[Achernar]

I think the limit part was because of the innkeeper's L'Hospitality.

[sturmhauke]

Oh god, calculus jokes. Run away!

Look, the riddle doesn't have anything to do with limits or trig or fractions. Think of it this way: the amount that the three guests paid is a negative amount of money, while the amout that the attendant has is a positive amount of money. The riddle tricks you into thinking that the amount the guests paid is positive also. Put another way, you can't add money someone doesn't have to money someone else does have and get a number that makes any sense.

[Philster]

Look, if a chicken and a half, rents a room and a half, for a dollar and a half, how long will it take a monkey with a wooden leg to kick all the seeds out of a dill pickle?

[Mangetout]

Um... four.

[NurseCarmen]

The crux of the riddle lies in trying to find a room in New York City for $30 a night, much less $25. It is due to this hurdle that the greatest scientific minds of the world have been unable to come up with an acceptable answer.

[RiverRunner]

Originally posted by Philster
Look, if a chicken and a half, rents a room and a half, for a dollar and a half, how long will it take a monkey with a wooden leg to kick all the seeds out of a dill pickle?

A toothbrush, 'cause motorcycles don't have seat belts.


RR

[Liberal]

Now if each paid $10 and each got a $1 refund that equals $9 that each person paid for that 1 room.Right. $30 - $3 = $27.

The attendant got a two dollar tip.Right. And $27 - $2 = $25, which is the room's actual cost.

$27 plus $2 equals $29.Yes, but so what? They didn't tip the $2 from the $29. They tipped the $2 from the $5 refund.

What happened to the other dollar?There is no "other dollar". $2 + $3 + $25 = $30

[Chronos]

So, the other day, I had ten bucks in my pocket, and a friend asked to borrow a five. Well, then I had $5, and he had $5, and $5 - $5 = $0! All of my money dissappeared! Those mathematicians stole it from me! :mad:

[Monalisa9]

The #9 is weird, because, for one thing, in any multiple of 9, the numbers in it will add up to nine or a multiple thereof: for example...in 81, 8 and 1 are 9...in 63, 6 and 3 add up to 9...in 25,191,546,381, those numbers add up to 45, a multiple of 9.

[barbitu8]

Originally posted by Monalisa9
The #9 is weird, because, for one thing, in any multiple of 9, the numbers in it will add up to nine or a multiple thereof: for example...in 81, 8 and 1 are 9...in 63, 6 and 3 add up to 9...in 25,191,546,381, those numbers add up to 45, a multiple of 9. What's so weird about that? The same thing is true for "3." The "9" is just a special case of the "3" rule.

[Speaker for the Dead]

Er... 3 + 0 = 3. Good so far. 6 + 0 = 6. Well, maybe its an exception. 9 + 0 = 9. Er, 2 exceptions. 1 + 2 = 3. Well, that's a gooder. 1 + 5 = 6. *sighs*

Maybe I misunderstood you?

[Robot Arm]

Originally posted by Philster
Look, if a chicken and a half, rents a room and a half, for a dollar and a half, how long will it take a monkey with a wooden leg to kick all the seeds out of a dill pickle?
It doesn't matter, 'cause we're doing a money trick.

[Wikkit]

Originally posted by Speaker for the Dead
Er... 3 + 0 = 3. Good so far. 6 + 0 = 6. Well, maybe its an exception. 9 + 0 = 9. Er, 2 exceptions. 1 + 2 = 3. Well, that's a gooder. 1 + 5 = 6. *sighs*

Maybe I misunderstood you? Yes, you must have, if you're talking about the post above yours. If a number's digits are added up, and the sum is evenly divisible by 3, the number is divisble by three. The same is true for 9.

30: 3+0=3, evenly divisible by 3.
60: 6+0=6, evenly divisble by 3.
You can even extend it by adding the sum's digits.
23985940303212: 2+3+9+8+5+9+4+0+3+0+3+2+1+2=51, 5+1=6, evenly divisble by 3.

The real question is why Monalisa9 reincarnated one of the most annoying types of threads on the board. Going to hunt some grys later?

[samarm]

I think this (http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=138305) more recent thread may have prompted Monalisa9 to bump this old thread. Don't know why the comment about the number 9 wasn't just posted in the more recent thread, though.

[samarm]

Come to think of it, Monalisa9's post about the number 9 came straight out of the left field - it seems to be completely unrelated to the thread topic. :confused:

So anyway, what the heck happened to the missing dollar in the end? :)

[barbitu8]

Originally posted by samarm
So anyway, what the heck happened to the missing dollar in the end? :) Harry S. Truman got it.

[Boxcar]

Here is the correct answer

You have to devide the cost of the room by 3 which gives you $8.3333333 apiece x 3 = 28 plus the 2 for the bell boy.
The extra dollar went towards paying for the hotel and was devided by 3 which is 33.33333.

[yoyo3500]

I wished ANYthing in NYC could cost only $10....

[KarmaComa]

Originally posted by siva
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 in case anyone's still wondering about this: the 'riddle' comes from the fact that 27+2 is a completely irrelevant figure. The cost of the room is 27-2, and the total money kicking around is 27-2+5, i.e. the cost of the room minus -5, the cost of the refund.

Mind you, given the events of 9/11, I must say that every character in this riddle is a hero.
:smack:

[TalkAboutIslam]

ooooh it was so cute....

the explanation was fine too

[mikan]

Prediction: this thread will still be alive in the year 2005.

[barbitu8]

Originally posted by mikan
Prediction: this thread will still be alive in the year 2005. Yeah, even though it has been answered correctly several times and incorrectly numerous times.

[TalkAboutIslam]

Originally posted by barbitu8
Yeah, even though it has been answered correctly several times and incorrectly numerous times.

that is how forums work....

;)

[barbitu8]

Originally posted by TalkAboutIslam
that is how forums work....

;) You mean that after a question has been correctly answered in GQ it is considered proper for people to continue answering it incorrectly? I'll be darned.

[TalkAboutIslam]

no barbit

I meant people much too often do not read the whole thread .... they just read the intro, the last few posts, and then they ask :

what is all that about here ?????

[aahala]

The real riddle is where does one find a hotel dude that will refund $5.

[sweepkick]

Bah! Off-the-cuff explainations, algabraic equations, and in-depth analysis of this problem are for those with a dreadful lack of ambition. What *I* want to know is how I can incorporate the fundamentals of the missing dollar into my next money-making scheme.

[TalkAboutIslam]

Sweepkick .... you are real ambitious man..... lol

[Kaitlyn]

Originally posted by Wikkit
Yes, you must have, if you're talking about the post above yours. If a number's digits are added up, and the sum is evenly divisible by 3, the number is divisble by three. The same is true for 9.

30: 3+0=3, evenly divisible by 3.
60: 6+0=6, evenly divisble by 3.
You can even extend it by adding the sum's digits.
23985940303212: 2+3+9+8+5+9+4+0+3+0+3+2+1+2=51, 5+1=6, evenly divisble by 3.



True, but there is a difference between the two. Adding the digits for multiples of nine until you have a single digit always reduces to exactly nine. Multiples of three can reduce to 3, 6, or 9.

The divisibility formula that's really freaky is divisibility by 7. Take a multidigit number. Delete the last digit, double it, then subtract the result from the remaining digits. If the result is evenly divisible by 7 , the original number was also. This method sometimes reduces to 0, -7, or -14, but those are evenly divisible by seven, so the rule holds.

For example: 4333. Take off the final 3, double it, and subtract that from the remaining digits: 433-6=427. Repeating, take off the seven, double it, and subtract: 42-14=28. You now have a multple of 7, so the original 4333 is a multiple.

[TalkAboutIslam]

wow. you are a mathmatician Number Six ??

[Kaitlyn]

I taught middle school pre-algebra for two years (along with seventh and eighth grade English), but no, I'm not a mathemetician. I actually learned that trick for divisibility by 7 in 5th grade, and it's stuck with me ever since, just 'cause it seems so cool.

[sailor]

The rule of nines is just a particular case as in base n it will work with (n-1).

That rule of divisibility by seven can be proved by proving
Mod7(10*n+m) = Mod7(n-2*m)
Unfortunately I forgot what I knew of modular math. Can someone do it?

[TalkAboutIslam]

Originally posted by Number Six
I taught middle school pre-algebra for two years (along with seventh and eighth grade English), but no, I'm not a mathemetician. I actually learned that trick for divisibility by 7 in 5th grade, and it's stuck with me ever since, just 'cause it seems so cool.

it is really so cool for a rule and I am really impressed you still remember that


Teaching is one of the best ways of learning something, is not it ???

[Kaitlyn]

Teaching something to another is one of the best ways of reinforcing something you've learned, and reinforcing knowledge is one of the steps in learning, so yes.

[TalkAboutIslam]

Originally posted by Number Six
Teaching something to another is one of the best ways of reinforcing something you've learned, and reinforcing knowledge is one of the steps in learning, so yes.

well, well said

that is actually what I meant to say, but of course, you have the right expressions.....

all the difference bt a mother and a non-mother tongue speakers ....