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2022/09/08 14:40:50
First edition 2016/06/25

The missing dollar riddle is as mysterious as the two envelope paradox.

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.

The missing dollar riddle had been discussed in a famous paper about the two envelope paradox.

When I studied a famous paper on the two envelope paradox, I found a discussion about the missing dollar riddle.
I had not been satisfied with the explanation in the paper.So I tried to resolve the riddle.
And as the result of it, I found that the riddle is as mysterious as the two envelope paradox.

The missing dollar riddle

Using some articles including the article "Missing dollar riddle" (revision at 17:08, 8 June 2016) in the English language Wikipedia as reference, I summarized the riddle as follows.

↓ Revised on April 9, 2017.


  • The charge of the hotel is $10 per person.
  • A group of three guests had paid $30 as the fee.
  • The hotel decided to return $5 to the group of the guests.
  • The bellhop was given 5 one dollar bills to return to the guests.
  • He could not divide 5 bills by 3, so he kept 2 bills as a tip for himself.
  • Therefore the final payment by the guests was $27. And the bellhop kept $2 as a tip for him.
  • Finally the total payment of the guests was $29. But the first payment was $30.
  • Where had $1 gone?

A very simple answer to this riddle makes it even more mysterious

(Added on September 08, 2022)

The followings is a very simple answer to this riddle.
The calculation in the riddle has flaws as follows.
  • He could not divide 5 bills by 3, so he kept 2 bills as a tip for himself.
  • Therefore the final payment by the guests was $27. And the bellhop kept $2 as a tip.
  • Finally the total payment of the guests was $29.
    Whoa, the final payment $27 already contains the amount for the tip. So, we shouldn't add $2 to $27.
  • Where had $1 gone?
    Whoa, where had the returned $3 gone?

Consequently, this flawsome calculation should be fixed as follows.
  • He could not divide 5 bills by 3, so he kept 2 bills as a tip for himself.
  • Therefore the final payment by the guests was $27 which contains the final hotel fee $25 and the tip $2.
  • The sum of $27 and the returned $3 equals $30 which is matching the first payment.
    So no dollar was disappeared.
    Good!

There remain a mystery that for many people, myself included, it is very hard to notice the answer despite the answer is very simple.

Strangeness of the wording of this riddle

This paragraph was added on July 9, 2016.

A strangeness

The phrase "Therefore the final payment by the guests" suggests that the problem in the riddle is about accounting.
On the other hand, the phrase "Where had $1 gone?" suggests that the problem in the riddle is about the preservation of the quantity.
This means that there is a trick which induces swapping the issue.

One more strangeness

(Added on July 18, 2016.)

The phrase "$27 for the hotel fee" is strange because it is uncertain that the hotel held all of the $27.
And the phrase "$2 for the tip" is strange because the guests did not know about it.

So I got a hypothesis.
If these equivocal words are replaced by decent words then no paradox will occur.
I tested it.
  • He returned 3 one dollar bills to the guests and kept 2 one dollar bills as a tip for him.
  • The final payment by the guests was $27.
  • The $2 for the tip was a payment by somebody.
  • Finally the total payment was $29. But the first payment by the guests was $30.
  • Where had $1 come from?
     
  • Wait! What was the $29?  Does it have meaning?
  • This is not a riddle but a wrong calculation.

One more another strangeness

(Added on April 9, 2017)

The word "final payment" is ambiguus. There is a possibility that the meaning of it has been replaced as below.

statement meaning of the word "final payment"
The final payment by the guests was $27. spending
The bellhop kept $2 as a tip for him. hotel fee

Goal of this riddle

This paragraph was added on Jun 29, 2016.

The real nature of the calculation of the missing dollar

We can uncover real nature of the calculation by replacing equivocal words by mathematically clear terms as follows.

$1 = (the amount of money thought first outgoings) − (the amount of money thought final outgoings)
= (first hotel fee) − ((the amount of money thought final hotel fee) + (the amount of money thought the tip))
= (first hotel fee) − (((the amount of money thought first hotel fee) − (the amount of money thought the discount)) + (the tip))
= (first hotel fee) − (((first hotel fee) − (amount of money of the returned money)) + (the tip))
= (the returned money) − (the tip).

In this way, the real nature of the calculation is only calculating the difference of the returned money and the tip.

Anotner way to find it

This paragraph was added on July 2, 2016.

Let's condider the location of the money in the riddle.
On the following table, each number denote amount of money after the events
event   guests hotel bellhop all of them
start   30
(first)
    30
(first)
first payment     30
(first)
  30
(first)
discount in the riddle
(A)
0 27
(first − return)
2
(tip)
29
(first + (tip − return))
correct calculation
(B)
3
(return)
25
(first − (return + tip))
2
(tip)
30
(first)
difference
(A − B)
− return tip   tip − return

Verification

This my opinion predicts as follows.
If the bellhop kept $3 as a tip for him and returned $2 to the guest, then a magical $1 will come from nowhere.
I tested it.
  • He returned 2 one dollar bills to the guests and kept 3 one dollar bills as a tip for him.
  • The final payment by the guests was as follows.
    • $28 for the hotel fee
    • $3 for the tip
  • Finally the total payment of the guests was $31. But the first payment was $30.
  • Where had $1 come from?

The goal of this riddle

Why do we mistake the difference of the returned money and the tip for the difference of the first quantity and the final quantity?  ← Revisedon July 9, 2016.)
The goal of this riddle is to find the cause of this mistake.

Fallacies and resolutions

This paragraph was greatly revised on Jun 27, 2016.

Fallacy 1: Ignored setoff among the real tip and the true but intangible discount

The tip for the bellhop was interpreted as a payment.
But in fact the tip was the outgoings from the true discount ($5).
So we should not use fake discount ($3) to calculate the final payment (to hotel and bellhop).

My resolution for the ignored setoff

Resolution

I think that the cause of the fallacy is the confusion of tangible but fake incomings ($3) and true but intangible incomings ($5).
(The guests could touch the tangible incomings ($3). But they could not touch the true but intangible incomings ($5) and outgoings ($2).)

Verification 1

This my opinion predicts as follows.
If the true discount and the tip are tangible, then the magical money will derive no riddle.
I tested it.
  • He showed the 5 one dollar bills to the guests.
  • After the agreement of the guests, he returned 3 one dollar bills to the guests and kept 2 one dollar bills as a tip for him.
  • The final payment by the guests was as follows.
    • $25 for the hotel fee
    • $2 for the tip
  • Finally the total payment of the guests was $27. But the first payment was $30.
  • Where had $3 gone?
  • Surely into pockets of the guests.

Verification 2

This my opinion predicts as follows.
If the true discount and the tip are both intangible, then the magical money will derive no riddle.
I tested it.
  • He kept 5 bills as a tip for himself.
  • The final payment by the guests was as follows.
    • $30 for the hotel fee
    • $5 for the tip
  • Finally the total payment of the guests was $35. But the first payment was $30.
  • Where had $5 come from?
  • Surely from pocket of the bellhop.

Verification 3

This my opinion predicts as follows.
If the true discount is tangible and the tip is nothing, then the magical money will derive no riddle.
I tested it.
  • He return $5 to the leader of the group of the guests.
  • The final payment by the guests was as follows.
    • $25 for the hotel fee
    • $0 for the tip
  • Finally the total payment of the guests was $25. But the first payment was $30.
  • Where had $5 gone?
  • Surely into pocket of the leader.

My another resolution for the ignored setoff

This resolution was added on July 2, 2016.

Resolution

I think that the invisibility of amount in arrear of the hotel may be the cause of the fallacy.
(The guests could touch the tangible incomings ($3). But they could not imagine the intangible amount in arrear ($2) of the hotel.

Verification

The above verification 2 and 3 verify this new resolution too.  (← Revised on July 9, 2016.)
So I cannot decide which resolution I should take.

But when I think of the phrase "one dollar bills as a tip for him.", it is natural to think that $2 came from the group of guests. 
So I would like to withdraw this new resolution.
(↑ Added on July 9, 2016.)

My yet another resolution for the ignored setoff

(This resolution was added on July 24, 2016.)

Resolution

I think that the bellhop had different two roles during the calculation of magical money as follows.

calculation role of the bellhop
$27 for the hotel fee agent of the hotel
$2 for the tip individual person

And I think that it was one of the cause of the ignored setoff.

Verification

This my opinion predicts as follows.
If the role of the bellhop as the individual person was explicit then the magical money will derive no riddle.
I tested it.
  • He could not divide 5 bills by 3, so he returned 3 one dollar bills to the guests and kept 2 one dollar bills as a tip for him.
  • The guests gave him these 3 one dollar bills as a tip.
  • Therefore the final payment by the guests was as follows.
    • $27 for the hotel fee
    • $3 for the tip (with will of the guests)
  • Finally the total payment of the guests was $30. And the first payment was $30.
  • Balanced! But the amount of money in the pocket of the bellhop was $5.
  • Where had $2 (that was $5 − $3) come from?
I would like to withdraw this new resolution too.

My new resolution for the ignored setoff

(This resolution was added on February 7, 2017.)

Resolution

I think that the unusual process to tip the bellhop is the cause of the ignored setoff.

Verification

This my opinion predicts as follows.
Usual process to tip the bellhop will not make a riddle.
I tested it.
  • The bellhop returned 5 one dollar bills to the guests.
  • The guests could not divide 5 bills by 3, so they gave 2 one dollar bills to the bellhop as a tip for him.
  • Therefore the final payment by the guests was as follows.
    • $25 for the hotel fee
    • $2 for the tip (with will of the guests)
  • Finally the total payment of the guests was $27. And the first payment was $30.
  • Where had $3 gone?
  • Surely into pockets of the guests.

My another new resolution for the ignored setoff

(This resolution was added on April 8, 2017.)

Resolution

I think that simply the ambiguity of the word "final payment" is the cause of the ignored setoff.
The word "final payment" has two meanings as below. The difference of these amounts is $3 which is equal to the ignored setoff.

Verification

This my opinion predicts as follows.
If the ambiguity is erased then no riddle will be derived.
I tested it.
  • The bellhop could not divide 5 bills by 3, so he kept 2 bills as a tip for himself.
  • Therefore the final hotel fee was $25 and a tip for bellhop was $2.
  • First the outgoings of the guests was $30, and finally it was $27.
  • Where had $3 gone?
  • Surely into pockets of the guests.

Fallacy 2: Ignored incomings

This paragraph was added on Jun 26, 2016. The title was changed on July 2, 2016.

Why did the phrase "first payment" appear in the problem?
But in fact the calculation result $29 is "final payment" not "first payment".
So we should not compare $29 and $30.

Why did we forget the incomings ($3)?
It is unbelievable that we forget the money which the guests had just obtained..

My resolution for the ignored incomings

Resolution

I think that the cause of the fallacy is delicately close amount of money. (first payment $30 is delicately close to the final payment $29.)
It would let us mistake $29 for mutation of $30 and we would forget $3 which the guest had just obtained.

I think that this fallacy is most important.   So any explanation about the missing money riddle which does not refer to this fallacy is wrong.

Verification

This my opinion predicts as follows.
If the final payment is not close to the first payment, then the magical money will derive no riddle.
I tested it.
  • He returned 27 one dollar bills to the guests and kept 2 one dollar bills as a tip for him.
  • The final payment by the guests was as follows.
    • $3 for the hotel fee
    • $2 for the tip
  • Finally the total payment of the guests was $5. But the first payment was $30.
  • Where had $25 gone?
  • Surely into pockets of the guests.
I did one more test on July 10, 2016.
  • He returned 6 one dollar bills to the guests and kept 1 one dollar bill as a tip for him.
  • The final payment by the guests was as follows.
    • $24 for the hotel fee
    • $1 for the tip
  • Finally the total payment of the guests was $25. But the first payment was $30.
  • Where had $5 gone?
  • Surely into pockets of the guests.

My another resolution for the ignored incomings

This paragraph was added on July 2, 2016.

One of my another resolution is as follows. 

Resolution

The fallacy of the ignored setoff among the tip and the true discount may be one of the cause of the ignored incomings.

Verification

This my opinion predicts as follows.
If the setoff among the tip and the true discount is not ignored, then the magical money will derive no riddle.
I tested it.
  • He returned 3 one dollar bills to the guests and kept 2 one dollar bills as a tip for him.
  • The final payment by the guests was as follows.
    • $25 = $30 - $3 - $2 for the hotel fee
    • $2 for the tip
  • Finally the total payment of the guests was $27. But the first payment was $30.
  • Where had $3 gone?
  • Surely into pockets of the guests.

Wrong resolution for the ignored incomings

This paragraph was added on July 2, 2016. Revised on July 9, 2016

I think that following resolution is wrong.

Resolution

The fact that the amount of the returned money to the guests is small may be one of the cause of the ignored incomings.

Verification

This my opinion predicts as follows.
If large amount of money was returned to the guests, then the magical money will derive no riddle.
I tested it.
  • The charge of the hotel is $333 per person.
  • A group of three guests had paid $999 as the fee.
  • The hotel decided to return $499 to the group of the guests.
  • The bellhop was given 499 one dollar bills to return to the guests.
  • He returned 250 one dollar bills to the guests and kept 249 one dollar bills as a tip for him.
  • The final payment by the guests was as follows.
    • $749 for the hotel fee
    • $249 for the tip
  • Finally the total payment of the guests was $998. But the first payment was $999.
  • Where had $1 gone?
  • It is a riddle!
I guess that this resolution is wrong.

Where had $1 gone?

Revised on Jun 27, 2016.

Effect of the first fallacy

The ignored setoff among the real tip ($2) and the true discount ($5) made the earning of $2.
As the result of this fallacy, the $2 was double counted. (first in the ignored setoff, second in payment for the tip)  (← Revised on Jun 26, 2016)

Effect of the second fallacy

The forgotten incomings ($3) made the loss of $3.

Consequence

The magical $1 was difference of them.

account event outgoings incommings balance
the group of guests fallacy 1   2
(ignored setoff)
2
fallacy 2 3
(forgotten incomings)
  − 3
aggregation   3 2 2 − 3


Correct calculations – From point of view of accounting –

Revised on July 2, 2016.

Correct calculation 1

If we give priority to "$2 for the tip" over "$27 for the hotel fee", to calculate correctly, we need to think of intangible but true incomings and outgoings. (← Revised on July 9, 2016.)
The correct calculation will be as follows.

account event outgoings incommings
the group of guests first payment 30  
discount (not fake)   5
payment for tip 2  
the hotel first payment   30
discount (not fake) 5  
the bellhop keeping as a tip   2
all   37
Balanced!!!
37
Balanced!!!

Correct calculation 2

If we give priority to "$27 for the hotel fee" over "$2 for the tip", to calculate correctly, we need to think of invisible amount in arrear of the hotel. (← Revised on July 9, 2016.)
The correct calculation will be as follows.

account event outgoings incommings
the group of guests first payment 30  
discount (fake)   3
the hotel first payment   30
discount (not fake) 5  
invisible amount in arrear   2
payment for tip 2  
the bellhop keeping as a tip   2
all   37
Balanced!!!
37
Balanced!!!

Correct calculation 3

(Added on July 17, 2016.)

If we think the bellhop as a cash carrier, we can analyze simply.
The correct calculation will be as follows.

account event outgoings incommings
the group of guests first payment 30  
discount (fake)   3
the hotel first payment   30
discount (not fake)
and
entrusting with $5
5  
the bellhop entrusting with $5   5
handing $3 to the guests as a discount (fake)
and keeping $2 as a tip
3  
all   38
Balanced!!!
38
Balanced!!!

Correct calculation – From point of view of conservation of ammunt of money –

This paragraph was added on May 4, 2017.

If we look the problem from point of view of cnservation of money, the correct calculation will be as follows.

event the group of guests the hotel the bellhop
(employee)
the bellhop
(private person)
total
before
check in
30       30
check in   30     30
conserved
discount (not fake)
and
entrusting with $5
  25 5   30
conserved
keeping as a tip   25 3 2 30
conserved
cash back
to the guests
3 25   2 30
conserved


New riddle

This paragraph was added on July 16, 2016.

  • The charge of the hotel is $10 per person.
  • A group of three guests had paid $30 as the fee.
  • The hotel decided to return $5 to the group of the guests.
  • The bellhop was given 5 one dollar bills to return to the guests.
  • He could not divide 5 bills by 3, so he kept 2 bills as a tip for himself.
  • Therefore the final payment by the guests was as follows.
    • $27 for the hotel fee
    • $2 for the tip
  • Finally the total payment of the guests was $29. But the first payment was $30.
  • Where had $1 gone? (The original riddle)
     
  • Oh!   I have forgotten $3 which are in the pockets of the guests.
  • The final payment by the guests was as follows.
    • $27 for the hotel fee
    • $2 for the tip
  • Finally the total payment of the guests was $29 (That is $27 + $2). But the real payment was $27 (That is $30 − $3).
  • Where had $2 come from?
     
  • Oops!   I have forgotten the source of $2 which was given to the bellhop.
  • The 2 one doller bills must have been stolen from the hotel.
  • The final payment by the guests was as follows.
    • $27 for the hotel fee
    • $2 for the tip
    • − $2 (stolen from the hotel)
  • Finally the total payment of the guests was $27 (That is $27 + $2 − $2), and the real payment was $27 (That is $30 − $3).
  • Balanced!   But could the guests steal? (A new riddle!)



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