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# Where's the missing dollar?

## Where's the missing dollar?

(OP)
I was hesitant to ask this question earlier because it didn't really have a whole lot to do with programming.  But after the question about the lottery turned into more of a math discussion I figured this would be fair game.

Before I ask the question let me add that it doesn't really have an answer because the logic in the question is flawed.  When I was in high school I did get my physics teacher to blow a whole class period trying to figure it out though, so it's always stuck in my head

[The question]
3 businessmen are on a trip and share a hotel room for the evening.  The bill comes to 30 dollars (it's a cheap hotel) so each businessman pays his even share - 10 dollars.  They head up to their room and the cashier realizes he's made a mistake.  He forgot that there's a special rate for the evening and that he should have only charged the businessmen 25 dollars (it's a REALLY cheap hotel).  The cashier then instructs the bellboy to run up to the room and refund the 5 dollars to the businessmen.  With his incredible insight, the bellboy realizes that the businessmen cannot easily split the 5 dollars between the 3 of them.  So instead he pockets 2 dollars and returns 3 dollars to the businessmen.  The 3 dollars is returned and each businessman gets a dollar back.

At this point each businessman has payed 9 dollars out of his own pocket, making a grand total of 27 dollars paid for the room.  Combine that with the 2 dollars that the bellboy kept yields 29 dollars.  So, where's the missing dollar?

-kaht

<P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <.</B>
(He's back)

Replies continue below

### RE: Where's the missing dollar?

The Bell Boys money was included in the 27 they payed.  So you cannot add it again to the 27 dollars.

Each bussiness man did pay 10 dollars = 30
The actual cost of the room ended up at 25 = 5 dollars left
Bell boy keeps 2 and returns the other 3 = nothing left.

It was however, an interesting read. :)

Wow, i'm having amnesia and deja vu at the same time.
I think I've forgotten this before.

### RE: Where's the missing dollar?

#### Quote:

At this point each businessman has payed 9 dollars out of his own pocket, making a grand total of 27 dollars paid for the room.
That accounts for the "outgo" -- the "income" is shared by the hotel and the bellboy: 25 dollars to the hotel, 2 dollars to the boy, accounting for the total of 27 dollars.

### RE: Where's the missing dollar?

So in reality 1 guy paid an extra dollar to the room, and the other 2 tipped the bell boy

### RE: Where's the missing dollar?

I wouldn't call it tipping in this case.  The money wasn't freely given by the business men to the bellboy.

-George

Strong and bitter words indicate a weak cause. - Fortune cookie wisdom

### RE: Where's the missing dollar?

#### Quote:

So in reality 1 guy paid an extra dollar to the room, and the other 2 tipped the bell boy
See that's the trap of this puzzle. Each paid th same (9 dollars) and the bellboy stole two. 30 - 25 - 3 - 2 = 0

### RE: Where's the missing dollar?

The businessman paid $25, the bellhop gave back 3 dollars, so 25+3 =$28.  The bellhop kept $2, so$28+2 =  30

No missing dollar.

____________________________________
Just Imagine.
http://www.imajinarts.com

### RE: Where's the missing dollar?

GUJUm0deL,

Your reasoning is a little flawed I think. The Businessmen paid $27. The Frontdesk got$25 and the bellhop got $2.$25 + $2 =$27

All,

Here is an object-oriented approach to solving this in Visual FoxPro.

#### CODE

#DEFINE ROOMRATE 10 && per person
lnTotalCashWas = 0

*!* Before the businessmen walk into the hotel
*!* they each put $10 in their pocket FOR lnCount = 1 TO lnTotalBusinessmen colBusinessmen.Add(CREATEOBJECT("Person")) colBusinessmen.Item(lnCount).TakeCash(10) lnTotalCashWas = lnTotalCashWas + 10 ENDFOR *!* At the frontdesk paying for their rooms loFrontDesk = CREATEOBJECT("Person") FOR EACH loBusinessman IN colBusinessmen loFrontDesk.TakeCash(loBusinessman.GiveCash(ROOMRATE)) NEXT *!* When special rate is realized *!* Frontdesk gives bellhop$5 to give to
loBellhop = CREATEOBJECT("Person")
loBellhop.TakeCash(loFrontDesk.GiveCash(5))

*!* Up at the room(s) each businessman is given $1 *!* by the bellhop FOR EACH loBusinessman IN colBusinessmen loBusinessman.TakeCash(loBellhop.GiveCash(1)) NEXT *!* Check to make sure that total still is$30
lnTotalCashIs = 0
NEXT
CLEAR
? "Businessmen Cash: " + TRANSFORM(lnTotalCashIs)
lnTotalCashIs = lnTotalCashIs + loFrontDesk.CashOnHand
? "Frontdesk Cash: " + TRANSFORM(loFrontDesk.CashOnHand)
lnTotalCashIs = lnTotalCashIs + loBellhop.CashOnHand
? "Bellhop Cash: " + TRANSFORM(loBellhop.CashOnHand)
?
?"Total Cash Was: " + TRANSFORM(lnTotalCashWas)
?"Total Cash Is: " + TRANSFORM(lnTotalCashIs)
?
?"Are they Equal? " + IIF(lnTotalCashWas == lnTotalCashIs, "YES", "NO")

DEFINE CLASS Person as Custom
CashOnHand = 0
PROCEDURE GiveCash(tnAmount)
LOCAL lnReturn
lnReturn = 0
IF tnAmount <= This.CashOnHand
This.CashOnHand = This.CashOnHand - tnAmount
lnReturn = tnAmount
ENDIF
RETURN tnAmount
ENDPROC
PROCEDURE TakeCash(tnAmount)
This.CashOnHand = This.CashOnHand + tnAmount
ENDPROC
ENDDEFINE

### RE: Where's the missing dollar?

I don't know how many times this question took round the world. We had to answer the same question when we were in the school.(1970-1980)

________________________________________________________
Zameer Abdulla
Help to find Missing people
Sharp acids corrode their own containers.

### RE: Where's the missing dollar?

As others have said, the businessmen pay 27 dollars, the hotel gets 25 and the bellboy steals 2.

Imagine that the bellboy had stolen all five dollars - hard to explain why he didn't, in fact.  Then the 'false sum' doesn't work, they have each paid 10 dollars and the bellboy has 5, no 'missing dollar'.

All down to misdirection.

------------------------------
An old man who lives in the UK

### RE: Where's the missing dollar?

(OP)

#### Quote:

Then the 'false sum' doesn't work, they have each paid 10 dollars and the bellboy has 5, no 'missing dollar'.

Sure it does.

I'll replace the last line of the puzzle with your new numbers:

#### Quote:

At this point each businessman has payed 10 dollars out of his own pocket, making a grand total of 30 dollars paid for the room.  Combine that with the 5 dollars that the bellboy kept yields 35 dollars.  So, where's the missing 5 dollars?

Of course, in this scenario it's not as difficult because we know that it's in the bellboy's pocket but that's besides the point.

The main "trick" about this puzzle is to ignore the magic number 30.  We're not concerned with that figure anymore.  The ultimate number we're trying to get to now is 25, because that is now the price of the room.  And simple logic in either case shows us that 27 - 2 (or 30 - 5 using the modified numbers) is equal to the new magic number of 25.  The puzzle just yields a number so close to 30 that it confuses the readers into thinking that it's the number you're striving to achieve - when in fact it's 25.

-kaht

<P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <.</B>
(He's back)

### RE: Where's the missing dollar?

I like kaht's approach with the five but would like to modify it slightly.

At this point each businessman has payed 10 dollars out of his own pocket, making a grand total of 30 dollars paid for the room.  Combine that with the 5 dollars that the bellboy kept yields 35 dollars.  So, where's the missing 5 dollars?

SB

At this point each businessman has payed 10 dollars out of his own pocket, making a grand total of 30 dollars paid for the room.  Combine that with the 5 dollars that the bellboy kept yields 35 dollars.  So, where did the extra five dollars come from?

Wow, i'm having amnesia and deja vu at the same time.
I think I've forgotten this before.

### RE: Where's the missing dollar?

(OP)
Errr... yeah, that's what I meant

Too much copy/pasting from above, heh.

-kaht

<P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <B> <P> <.</B>
(He's back)

### RE: Where's the missing dollar?

I have a quick eye for spotting easy money, unfortunatly as in your example it's always missing in the end.

Wow, i'm having amnesia and deja vu at the same time.
I think I've forgotten this before.

### RE: Where's the missing dollar?

I remember this same puzzle in AsiaMagazine way back in the early 60s.

### RE: Where's the missing dollar?

This little brainteaser is making its rounds once again.

My son & I had the manager and over 100 people in a Bonanza restaurant ready to hang us last night.  They were yelling answers at us as we made a hasty retreat to the car.

All these posted "answers" remind me of 'a dog chasing his tail.' This age-old classic wouldn't still be alive if an answer was this easy.

Changing the way the original question is asked, or looking at it from the different characters' viewpoints is like changing the rules during a game because you're losing.

Face it...there is no answer when asked the original way.  That's why it's a classic!

### RE: Where's the missing dollar?

But the answer is easy, just non-obvious. Most of the answers here explain it quite happily, there's no disagreement (unlike the Monty Hall problem ...)

### RE: Where's the missing dollar?

(OP)
It's definitely a good way to blow off a lot of time if you tell it in a room of a bunch of people that have never heard it before.  Everybody has their own explanation.

-kaht

Looking for a puppy?

Silky Terriers are small, relatively odorless dogs that shed no fur and make great indoor pets.

### RE: Where's the missing dollar?

I guess that the saddest twist to this story in the 21st century is that the story-teller has to preface the story-riddle, explaining that "3 men in 1 motel room" has nothing to do with a perverted lifestyle.
Get past that and you can frustrate some of the deepest thinkers.

### RE: Where's the missing dollar?

(OP)
Ehh... I didn't say anything about it being perverted..... however, I did say it was a cheap hotel - so I guess that's pretty much the same thing

-kaht

Looking for a puppy?

Silky Terriers are small, relatively odorless dogs that shed no fur and make great indoor pets.

### RE: Where's the missing dollar?

I guess that the saddest twist to this story in the 21st century is that the story-teller has to preface the story-riddle, explaining that "3 men in 1 motel room" has nothing to do with a perverted lifestyle.

Just start it like this:
"Three very cheap businessmen are on a trip and, in order to save some money, share a very cheap hotel room for the evening."

### RE: Where's the missing dollar?

Or multiply all values by 10.

### RE: Where's the missing dollar?

Make it 300 dollars charged.  It should be 250.  The cashier takes back 50 dollars for a refund, but decided to keep 20 and give each of them ten.

Each businessman has paid 100 dollars and got 10 back, net payment of 90, times three equals 270.

The hotel got 250.  The cashier stole 20, that accounts for the 270.

Or if he'd refunded 48 dollars, 16 each, they would each have paid 84 dollars, total 252.  Adding this to the 2 dollars that this alternative cashier stole gives you 254, way out.  Adding the 250 dollars received by the hotel to the 2 stolen is also 252, no missing dollars.

------------------------------
An old man who lives in the UK

Is the $30 an hourly rate? --just wondering-- Gooser ### RE: Where's the missing dollar? I thought, Pgould1 had a problem not with the prices being low, but with the fact that "in the 21st century ... the story-teller has to preface the story-riddle, explaining that "3 men in 1 motel room" has nothing to do with a perverted lifestyle"? ### RE: Where's the missing dollar? (OP) Stella, I believe the humor lies in the fact that hotels with low prices are notorious for being used for only perverted purposes. -kaht Looking for a puppy? Silky Terriers are small, relatively odorless dogs that shed no fur and make great indoor pets. ### RE: Where's the missing dollar? I thinkt he only reason there's any disagreement about the Monty Hall problem is because of poor logicians, not because of poor logic. It's quite simple, really. :) ### RE: Where's the missing dollar? Oh oh ... ### RE: Where's the missing dollar? laugh Just use the "extremes" technique. By pushing the number to an extreme it simplifies the problem. There are a million boxes, one of which randomly contains a prize. You are allowed to choose one and do so. The contents of the remaining boxes you did not choose then get combined into one big box. Now, you can keep your original box (1 in a million chance) or you can switch to the big box (999,999 in a million chance). What do you do? A little thought shows that "combining n boxes into one big box" is the same as "removing n-1 boxes known to be empty." There's a gigantic wikipedia page on this which could really use a simpler explanation like the above. ### RE: Where's the missing dollar? Here we go ... ### RE: Where's the missing dollar? What does that have to do with the original puzzle? I'll take the big box. The answer is "42" ### RE: Where's the missing dollar? Sorry about the off-topic wandering. And I meant no disrespect for anyone! In fact, I'd be content if we all just forgot the off-topic parts of this thread... silly me. ### RE: Where's the missing dollar? No disrespect, just curious. The answer is "42" ### RE: Where's the missing dollar? That's all right, esquared...after all, strongm started it. As for the original topic: although I am NOT an accountant, nor do I pretend to any skill in the subject, it strikes me that it's the sort of thing that accountants deal with all the time. I would say that the puzzle is attempting to introduce confusion between assets and liabilities, debits and credits. The way to introduce the confusion is to suggest that the boy's$2 credit goes also to the hotel as a $2 credit, which is of course wrong. The fact that the boy is a representative of the hotel is a compelling reason to perceive them as one entity. It's also irrelevant...a master touch! So, in fact, the boy's$2 credit is of course a \$2 debit to the hotel.

So, the transactions appear to be:
1.  credit hotel 30, debit businessmen 30.
2.  debit hotel 5, credit bellboy 5.
3.  debit bellboy 3, credit businessmen 3.

Total assets to hotel, 25.  Total assets to bellboy, 2.  Totall assets overall, 27.  Total liabilities to businessmen, 27.

All in balance, all in harmony....thanks to Amatino Manucci.

Bob

### RE: Where's the missing dollar?

This riddle is older than me. (And that's saying a lot.)

Greg
"Personally, I am always ready to learn, although I do not always like being taught." - Winston Churchill

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