## The Case of the Missing Dollar

A forum for good logic/math puzzles.

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sgware
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### The Case of the Missing Dollar

Sorry if this has been posted before... I'm new to the forums (though old to the web comic). I did a brief search for 'dollar' and didn't find it, so if it's out there, it's in another form.

Three men on a business trip check into a motel for the night. The cost of a room is \$30, so they agree to split it evenly and each pay \$10. The manager decides that, since business is good, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a two dollar tip. They split the remaining three dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$9, and the bellhop has been given \$2. But 9 x 3 + 2 = 29. What happened to the thirtieth dollar?
One day, I hope to change the world. Now if only they would give me the source code...

Macbi
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### Re: The Case of the Missing Dollar

I'm sure it has been posted before.
Spoiler:
They each pay 9 dollars making \$27, \$2 goes to the bellhop, leaving \$25, which is how much the meal cost.
Indigo is a lie.
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jestingrabbit
Factoids are just Datas that haven't grown up yet
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### Re: The Case of the Missing Dollar

Yeah, I *think* its been done here before, but I can't put my finger on it.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Sana
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### Re: The Case of the Missing Dollar

The trick lies in the final calculation.
Spoiler:
There is no reason to add \$2 to \$9 * 3. The room cost them \$25, the tip cost them \$2. That's \$27 total cost or \$9 per person. They paid \$30 or \$10 per person and received change of \$3 or \$1 per person.

\$30 payment - \$5 change = \$25 payment.
\$25 payment + \$2 tip = \$27 payment.
\$27 payment / 3 people = \$9 payment / person.

Multiplying \$9 by 3 already yields the cost of the room plus the tip, so it makes no sense to add \$2. The only remaining money is their change of \$3.

Cosmologicon
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### Re: The Case of the Missing Dollar

sgware wrote:Three men on a business trip check into a motel for the night. The cost of a room is \$30, so they agree to split it evenly and each pay \$10. The manager decides that, since business is good, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a two dollar tip. They split the remaining three dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$9, and the bellhop has been given \$2. But 9 x 3 + 2 = 29. What happened to the thirtieth dollar?

Spoiler:
The error in logic for this old chestnut becomes easy to see if you rewrite the story as if one-dollar-bills didn't exist:
Three men split a room for \$30, so they each pay \$10. The manager refunds them \$5, and so hands a five-dollar bill to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a five dollar tip, and don't take any for themselves.

This means that, at the end of the night, each of the three men has paid \$10, and the bellhop has been given \$5. But 10 x 3 + 5 = 35. Where did these five magic dollars come from?

MrMoto
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Location: Québec

### Re: The Case of the Missing Dollar

This is a classic riddle. I told it a few days ago to a fellow after he tried to rip me off.

Fellow: That comes to \$7.40, please.
Me: Sure, here's \$20.15.
F: [handing back 5¢] I'll just take the dime so that I can give you back 50¢.
M: Well, you can take my dime, but that would make 70¢ in change.
F: No, it's 50¢.
M: I truly believe that it would make 70¢. Better yet, let me give you this nickel in my hand, that you might give me 75¢.
F: No, sixty cents minus a dime makes 50¢.
M: But sixty cents plus a dime makes 70¢.
F: You have to subtract. 60 - 10 = 50.
M: True as that calculation may be, I believe you're subtracting when you should be adding.
F: 60 - 10 = 50.
M: [taking back 10¢] Tell you what, just give me change for the \$20.
F: [gives \$12.60]
M: Thanks. By the way, could I trouble you to give me a quarter for these two dimes and a nickel? And would you like to hear a story?

A'Tuin
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### Re: The Case of the Missing Dollar

Spoiler:
final total=25
3*9=27(-2)=25

Xias
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### Re: The Case of the Missing Dollar

You're all wrong, that doesn't explain where the missing dollar is.

Spoiler:
The next night, two men on a business trip check into the same motel. The cost of a room is \$30, so they agree to split it evenly and each pay \$15. The manager decides that, since business is good, and since the night before he was generous, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a three dollar tip. They split the remaining two dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$14, and the bellhop has been given \$3. But 14 x 2 + 3 = 31.

sgware
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### Re: The Case of the Missing Dollar

Haha! Well done Xias! That's the best solution I've ever heard to the problem.
One day, I hope to change the world. Now if only they would give me the source code...

Sana
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### Re: The Case of the Missing Dollar

Xias wrote:You're all wrong, that doesn't explain where the missing dollar is.

Spoiler:
The next night, two men on a business trip check into the same motel. The cost of a room is \$30, so they agree to split it evenly and each pay \$15. The manager decides that, since business is good, and since the night before he was generous, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a three dollar tip. They split the remaining two dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$14, and the bellhop has been given \$3. But 14 x 2 + 3 = 31.

Since when can you not split five dollars evenly? Well, I suppose if they didn't want to be bothered converting the dollar bills. Anyways, you are creating more problems with faulty mathematics. Joke or not.

Spoiler:
There is no reason to add \$3 to \$14 * 2. The room cost is \$25, the tip cost them \$3. That's \$28 total or \$14 per person. They paid \$30 or \$15 per person and received change of \$2 or \$1 per person.

\$30 payment - \$5 change = \$25 payment.
\$25 payment + \$3 tip = \$28 payment.
\$28 payment / 2 people = \$14 payment / person.

Multiplying \$14 by 2 already yields the cost of the room plus the tip, so it makes no sense to add \$3. The only remaining money is their change of \$2.

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### Re: The Case of the Missing Dollar

Each paid 25/3 dollars for the room and each paid the bellhop 2/3 of a dollar tip. word to your mom.

Token
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### Re: The Case of the Missing Dollar

Sana wrote:
Xias wrote:You're all wrong, that doesn't explain where the missing dollar is.

Spoiler:
The next night, two men on a business trip check into the same motel. The cost of a room is \$30, so they agree to split it evenly and each pay \$15. The manager decides that, since business is good, and since the night before he was generous, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a three dollar tip. They split the remaining two dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$14, and the bellhop has been given \$3. But 14 x 2 + 3 = 31.

Since when can you not split five dollars evenly? Well, I suppose if they didn't want to be bothered converting the dollar bills. Anyways, you are creating more problems with faulty mathematics. Joke or not.

Spoiler:
There is no reason to add \$3 to \$14 * 2. The room cost is \$25, the tip cost them \$3. That's \$28 total or \$14 per person. They paid \$30 or \$15 per person and received change of \$2 or \$1 per person.

\$30 payment - \$5 change = \$25 payment.
\$25 payment + \$3 tip = \$28 payment.
\$28 payment / 2 people = \$14 payment / person.

Multiplying \$14 by 2 already yields the cost of the room plus the tip, so it makes no sense to add \$3. The only remaining money is their change of \$2.

Words fail me.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

Robin S
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### Re: The Case of the Missing Dollar

An elegant way (though I say so myself) of phrasing the solution is as follows:

Spoiler:
each of the three men has paid \$9, and the bellhop has been given \$2. But 9 x 3 + 2 = 29.
should read "each of the three men has paid \$9, from which the bellhop has been given \$2. 9 x 3 = 27 (don't count the \$2 twice) and the remaining \$3 resides with the businessmen."
This is a placeholder until I think of something more creative to put here.

Sana
Posts: 123
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### Re: The Case of the Missing Dollar

Token wrote:
Sana wrote:
Xias wrote:You're all wrong, that doesn't explain where the missing dollar is.

Spoiler:
The next night, two men on a business trip check into the same motel. The cost of a room is \$30, so they agree to split it evenly and each pay \$15. The manager decides that, since business is good, and since the night before he was generous, he will refund them \$5, and so hands five one dollar bills to the bellhop. Realizing that they cannot split \$5 evenly, the men give the bellhop a three dollar tip. They split the remaining two dollars evenly amongst themselves.

This means that, at the end of the night, each of the three men has paid \$14, and the bellhop has been given \$3. But 14 x 2 + 3 = 31.

Since when can you not split five dollars evenly? Well, I suppose if they didn't want to be bothered converting the dollar bills. Anyways, you are creating more problems with faulty mathematics. Joke or not.

Spoiler:
There is no reason to add \$3 to \$14 * 2. The room cost is \$25, the tip cost them \$3. That's \$28 total or \$14 per person. They paid \$30 or \$15 per person and received change of \$2 or \$1 per person.

\$30 payment - \$5 change = \$25 payment.
\$25 payment + \$3 tip = \$28 payment.
\$28 payment / 2 people = \$14 payment / person.

Multiplying \$14 by 2 already yields the cost of the room plus the tip, so it makes no sense to add \$3. The only remaining money is their change of \$2.

Words fail me.

It is flattering that I have left you speechless.

quintopia
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### Re: The Case of the Missing Dollar

When trying to come up with way to state this riddle in my own words, I find it hard to do so in a way that doesn't make the calculation error obvious. . .but would it be obvious to someone who has never heard the riddle? I wonder how the person who originally came up with it knew that it would confuse people?

Robin S
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### Re: The Case of the Missing Dollar

I had heard it long ago, but did not remember the solution. It was obvious there had been a calculation error of some sort, so I just went through each sentence of the riddle until I found it.
This is a placeholder until I think of something more creative to put here.

MrMoto
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### Re: The Case of the Missing Dollar

I just noticed that my location appears as "QuÃ©bec".
That's almost insulting.

Fixed. But in the future, you can ask for help in the Site / Forum Issues area. -Hodge.

teatea
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### Re: The Case of the Missing Dollar

None of you actually explained where the missing dollar really is. I think I have a solution that best fits the description.

Spoiler:
The Three men ended up paying a total of 3X9=\$27. That's the 25 plus the 2 stolen from the bellboy. The question tries to trick you into adding the 2 stolen from the bellboy to the total of 27 that the 3 business men paid, for a total of 29. But that's a load off bull "faeces". The 27 they end up paying was actually the 25 for the room and the 2 stolen from the bellboy. So the 27 they end up paying plus the 3 that is returned to them is the original 30. Thus the 25 plus the 5 that the bell boy was supposed to return to them equals the original 30.

A simpler way but less detailed...

Kept by the motel manager:\$25
stolen by the bell boy:\$2
The money that is returned:\$3
Total: \$30 As required

phlip
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### Re: The Case of the Missing Dollar

teatea wrote:None of you actually explained where the missing dollar really is.

Because Macbi, Sana, Cosmo, A'Tuin, Sana again and Robin S don't count.

Seriously, someone's given the solution in nearly half of the posts in this thread.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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nuggetmonkey
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### Re: The Case of the Missing Dollar

This guy took the dollar.

baf
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### Re: The Case of the Missing Dollar

This problem always reminds me of a trick from elementary school. "You have eleven fingers!" proclaims one child. "I can prove it. Count them." The other child obediently counts to ten on their fingers. "Now count them backward, but stop after one hand." The other child counts from ten down to six. "Plus five on the other hand is eleven!"

In general, you can produce whatever number you want if you don't care about what it actually means.

Nexus_1101
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### Re: The Case of the Missing Dollar

it a cse of math and correct calculation

Spoiler:
when calculating 9*3 you donot ad the two that the bell hop kept you add the \$3 that the men had returned
therfor:
9*3+3 = 30
Ahh it all adds up now!!!
HOSTING -> Heroquest
CHAOS BONUS

Code: Select all

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LE4dGOLEM
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### Re: The Case of the Missing Dollar

baf wrote:This problem always reminds me of a trick from elementary school. "You have eleven fingers!" proclaims one child. "I can prove it. Count them." The other child obediently counts to ten on their fingers. "Now count them backward, but stop after one hand." The other child counts from ten down to six. "Plus five on the other hand is eleven!"

In general, you can produce whatever number you want if you don't care about what it actually means.

It took me until well into highschool to figure out that the best response is to call them a "filfthy inbred".
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Sykotic1189
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### Re: The Case of the Missing Dollar

The problem with every solution so far, except Xias's because it uses the same flawed logic so I would hope he gets it too, is that everyone keeps saying you subtract the tip from the \$27 to get the \$25 for the room. The problem is you still don't get the original \$30 back. Then we have the case of the missing \$5.

Spoiler:
The problem is in the wording itself. We assume that since they each paid \$10 and got one back they all paid \$9. Truth is that if they had paid the \$25 at the desk two would have paid \$8 and the third would have paid \$9. So when the bellboy only gave each one \$1 back they actually paid \$28 with a \$9,\$9,\$10 split. Add the bellboy's tip in, you get the original \$30.

Edit: just realized how very late this is, but I cannot accept flawed logic and flawed math.
Good times?

I'm not paranoid, I sleep with a 14'' boot knife for other reasons.

Cauchy
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### Re: The Case of the Missing Dollar

I'm pretty bad at detesting internet sarcasm, so let me just clarify: that last post as a joke, right?
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

Macbi
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### Re: The Case of the Missing Dollar

Cauchy wrote:I'm pretty bad at detesting internet sarcasm, so let me just clarify: that last post as a joke, right?

That post makes perfect sense, it just doesn't answer the problem. Sykotic1189, all you've done is explain (in a roundabout way) where all the money goes. The only correct way to "answer" the problem is to show the error in the calculations leading to the missing dollar. Merely doing your own calculations doesn't show what's up.
Indigo is a lie.
Which idiot decided that websites can't go within 4cm of the edge of the screen?
There should be a null word, for the question "Is anybody there?" and to see if microphones are on.

Lord Aurora
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### Re: The Case of the Missing Dollar

Cauchy wrote:I'm pretty bad at detesting internet sarcasm, so let me just clarify: that last post as a joke, right?

I find it quite easy to detest anything I find on the internet.
Decker wrote:Children! Children! There's no need to fight. You're ALL stupid.

Sykotic1189
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### Re: The Case of the Missing Dollar

I tried to say it was in how the story is told, but I lack a certain skill in explaining my thoughts or my math but I'm gonna try to make it simple.

Spoiler:
\$30(room)/3(people)=\$10(per person)

Clerk finds mistake.

\$25(new room price)/3(people)=\$8.33(per person)

Since everybody gets \$1 and 2\$ goes missing, grand total they each paid \$9.33 for the room.

\$9.33*3=\$28+\$2=\$30.

The dollar disappears due to a rounding error in the money because they only use whole units. Using whole units, they didn't actually pay evenly for the room. Two guys paid \$8 and the third paid \$9.

\$8+\$8+\$9=\$25

I really can't make it much simpler.
Good times?

I'm not paranoid, I sleep with a 14'' boot knife for other reasons.

phlip
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### Re: The Case of the Missing Dollar

(I too, am not sure if Sykotic is joking... if so, then ignore me.)

Though you get \$30 at the end, that really doesn't work... mostly because the people didn't pay \$9, \$9 and \$10... all three walk away from the experience exactly \$9 poorer. There's no rounding error here.

The \$30 is \$25 for the room, plus \$2 for the tip, plus \$3 returned as change. The net amount paid is \$25 + \$2 = \$27, paid as \$9 each... add \$3 for the change, and you get the \$30 back.

But you're saying that the net amount they paid is \$25 + \$3 = \$28, and then adding in the other \$2 later. So you get the same result of \$30 (since you're just doing the additions in a different order), but the \$28 figure for "what they actually paid" is still wrong.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

Sana
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### Re: The Case of the Missing Dollar

Sykotic1189 wrote:I tried to say it was in how the story is told, but I lack a certain skill in explaining my thoughts or my math but I'm gonna try to make it simple.

Spoiler:
\$30(room)/3(people)=\$10(per person)

Clerk finds mistake.

\$25(new room price)/3(people)=\$8.33(per person)

Since everybody gets \$1 and 2\$ goes missing, grand total they each paid \$9.33 for the room.

\$9.33*3=\$28+\$2=\$30.

The dollar disappears due to a rounding error in the money because they only use whole units. Using whole units, they didn't actually pay evenly for the room. Two guys paid \$8 and the third paid \$9.

\$8+\$8+\$9=\$25

I really can't make it much simpler.

Spoiler:
They each paid \$9. The room costs \$25. The bellhop takes \$2 tip for himself. Nothing is missing, as 9 * 3 = 27

Sykotic1189
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### Re: The Case of the Missing Dollar

I think my problem was that I was trying to get the original \$30 back, and I am too proud to admit mistake.
Good times?

I'm not paranoid, I sleep with a 14'' boot knife for other reasons.

Sana
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### Re: The Case of the Missing Dollar

Receipt:

Spoiler:
Cost of room: \$25, tip: \$2, cash tendered: \$30, change: \$3.

Xias
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### Re: The Case of the Missing Dollar

Sykotic1189 wrote:The problem with every solution so far, except Xias's because it uses the same flawed logic so I would hope he gets it too, is that everyone keeps saying you subtract the tip from the \$27 to get the \$25 for the room. The problem is you still don't get the original \$30 back. Then we have the case of the missing \$5.

Spoiler:
The problem is in the wording itself. We assume that since they each paid \$10 and got one back they all paid \$9. Truth is that if they had paid the \$25 at the desk two would have paid \$8 and the third would have paid \$9. So when the bellboy only gave each one \$1 back they actually paid \$28 with a \$9,\$9,\$10 split. Add the bellboy's tip in, you get the original \$30.

Edit: just realized how very late this is, but I cannot accept flawed logic and flawed math.

I believe yours is still flawed.
Spoiler:
The men DID spend \$27 total. They gave \$30, got \$3 back, therefore the total is \$27, split between them is \$9 each. That's not where the error is. The error is in adding the \$2 to the \$27, not in the values themselves.

Here is why:

Spoiler:
When the bellboy gives them each \$1 back, they each paid \$9. The puzzle is correct up to that point. I am not sure how you figure they spent a total of \$28; if they each spent \$10 and got \$1 back they each spent \$9.

The problem with the puzzle and your logic is that both are adding money from two different sets together. One set being "money spent" and the other being "money held at the end".

A: Money Spent:
Av: Men: \$27
Ab: Bellboy: \$3
Am: Manager: \$0 (you would think 5, but if you think about the \$30 that the men spend originally just being divided into \$25 for the manager and \$5 for the bellboy (which is logical) it makes sense.)
Total money spent: \$30

B: Money held at the end:
Bv: Men: \$3
Bb: Bellboy: \$2
Bm: Manager: \$25
Total money held: \$30

So, the sum of all the parts of each set add up to 30.

Now, several expressions can be made:
Av=Bm+Bb (The money the men spent is divided between the manager and the bellboy
Ab=Bv (The money the bellboy spent is given to the men)
Bv=Ab (" ")
Bb=Av-Bm (The money the bellboy pockets is the difference between the money the men spend and the money the manager keeps)
Bm= Av-Bb(The money the manager keeps is the difference between the money the men spend and the money the bellboy pockets)

Essentially, there are only two expressions, and the others are the same.

Here's where the problem in the story is. In the end, the expression they are trying to make is:

Av+Bb=A(=B=30)

As you can see, since you cannot use any of the expressions created to solve this, it is not true. You are trying to add
parts of two different sets to get the total for one set. That's where the error is.

tricky77puzzle
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### Re: The Case of the Missing Dollar

I actually got this the first time I heard it when I was in grade 3 and found it in a MENSA puzzle book.

I won't bother posting a solution since everyone else has already done so.

cybersleuther
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### Re: The Case of the Missing Dollar

A light extension, though belated (sorry - I'm new):-

Continuing on from the story in the original post...

Their wonderful night's sleep was followed the very next day by considerable discomfort from itchy welts, which, it was determined, resulted from bedbug bites they'd received during their stay at the motel.

Upon referring the matter to the manager they were each reimbursed their net \$9 fee (ie, their original \$10 less the \$1 goodwill discount they'd already been given). The bellhop, of course, retained the \$2 tip.

The final, pocketed money in this transaction: 3 x \$10 = \$30, + the bellhop's \$2 = \$32

In other words, the motel was paid \$30, returned it…and the bellhop is left with the 'spare' \$2 that materialised (a "cash float"?)

…???

Carnildo
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### Re: The Case of the Missing Dollar

That's too easy:
Spoiler:
The manager refunded them too much: they paid \$30, got \$5 returned that evening, and got \$27 returned the next morning.

cybersleuther
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### Re: The Case of the Missing Dollar

Carnildo wrote:That's too easy:

True. It's only a lightweight diversion, and lacks the subtlety of the original.

I came across the first part of the puzzle many years ago under the title, "The Missing Penny". It had a different storyline: about buying something from a shop, I think.

poirelli
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### Re: The Case of the Missing Dollar

Sana wrote:Since when can you not split five dollars evenly? Well, I suppose if they didn't want to be bothered converting the dollar bills. Anyways, you are creating more problems with faulty mathematics. Joke or not.

Since 500 (pennies) wasn't evenly divisible by 3. Which I think has been true since... forever.

pistolero
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### Re: The Case of the Missing Dollar

The problem makes more sense assuming one person pays for all of it.. man pays 10\$ a room for 3 rooms, is given 5\$ back.(has 5\$, hotel has 25) He gives bellhop a 2\$ tip(has 3\$, hotel has 25\$, bellhop has 2\$.) added together, it is 30.btw, he pays 10\$ a room before getting 5\$ back, 8.33 dollars a room after getting 5\$ back. [3x8.33(paid per room=25\$)+3(cash in hand)+2(bellhop tip)]=30. There is no missing dollar, it is just incorrect to assume that he paid 9\$ per room. A better action for the customer would have been to keep 2\$ and give the waiter 3, so he pays 27\$ (9\$ per room) instead of keeping 3, and giving away 2, which when added to 25 does not make a multiple of 3.

tl;dr: The missing dollar is that you said he pays 9\$ per room , when it is actually 8.33 per room after subtracting the 2\$ tip, and the 3\$ cash in hand.

EDIT: I've seen a lot of attempts, but I think mine is the only solution

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### Re: The Case of the Missing Dollar

pistolero wrote:it is just incorrect to assume that he paid 9\$ per room. A better action for the customer would have been to keep 2\$ and give the waiter 3, so he pays 27\$ (9\$ per room) instead of keeping 3, and giving away 2, which when added to 25 does not make a multiple of 3.

What? Sounds like you've got it backward. If a man pays \$30 for three rooms, then gets \$5 back, then gives away \$2, then net he pays 30 - 5 + 2 = \$27. If, as you suggest, he gives away \$3 instead, then net he pays 30 - 5 + 3 = \$28, not 27.

In your scenario, you're adding the 2 to the 25, instead of subtracting it from 30. This is exactly the same error as in the original problem statement.