Missing Dollar Riddle

[Reluctantly]

just wondering what people have to say about,

https://en.wikipedia.org/wiki/Missing_dollar_riddle

Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 as a tip for himself. Each guest got $1 back: so now each guest only paid $9; bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?​

 

[redbaron]

It's a trick of language that makes it appear like there's a paradox, but it's actually very simple if you just keep track of how much cash is with each individual party. The problem starts when it says, "Each guest has paid $9 for a total of $27". Which makes it look like the Hotel has $27 and the bellhop has $2 for a total of $29/30. It doesn't take into account the refund and theft.

With the refund and theft, the Hotel only took $25, while the bellhop took $2 of the $30. The other $3 were returned to the guests for a total of $30. The $27 + $2 is thrown in there deliberately to confuse people, when it's actually just the mathematics being applied incorrectly.

The easiest way I can think of displaying it is:
$30 are paid to Hotel.
$30 is split into $25 for Hotel and $5 for Bellhop.
$5 is then split into $3 for Guests and $2 for Bellhop.

Total: $25 Hotel plus $3 Guests plus $2 bellhop = $30.

 

[Reluctantly]

heh, yeah but that just tracks the money. If you track how much each person paid, it doesn't add up. Or does it?

 

[Cheeseumpuffs]

If you track how much each person paid, it doesn't add up. Or does it?

The customers paid $27, the hotel gained $25, and the bellhop gained $2.

-27 + 25 + 2 = 0

So yes, yes it does.

 

[Reluctantly]

right right, but if the bellhop gives 3 back, then each person is assumed to have paid $9 each for the room, at least from their point of view. $27+$2 = $29 paid for the room from the point of view of the bellhop and hotel guests. Can you explain why that dollar seems to have vanished in terms of what is paid for the room?

 

[Urakro]

heh, yeah but that just tracks the money. If you track how much each person paid, it doesn't add up. Or does it?

The $27 should never have been added with the $2. It's true the guests paid $27 all together after their $3 refund (30 - 3). The hotel only got $25 of that though. The remaining $2 was stolen.

With the above statement: $3 dollar refund + $25 for hotel room + $2 stolen.

Though I'll admit it did trick me too. First time I've seen that riddle. Good find!

 

[redbaron]

Adding the $27 and $2 is a mistake in the first place because the $2 the bellhop stole is a part of the $27 paid for the room by the guests, not an addition to it.

So after the refund the guests paid $27. $25 went to the hotel, $2 went to the Bellhop. No money missing.

From the point of view of the guests $27 is paid for the room.
From the point of view of the bellhop $25 is paid for the room and $2 for him.

It all still adds up to $27, then add $3 for the refund to get 30.

 

[Cheeseumpuffs]

$27+$2 = $29 paid for the room

No, the $2 is already counted in the $27. Adding $2 means you're counting the bellhop's money twice.

Can you explain why that dollar seems to have vanished in terms of what is paid for the room?

It's not that a dollar vanished, it's that you counted two twice and ignored the three.



It all even works out in the peoples' point of view.

From the customers' point of view, they paid $27 for a room and got $3 back
27+3=30

From the hotel's point of view, the customers paid $25 and got $5 back.
25+5=30

From the bellhop's point of view, the hotel gets $25, the customers get $3, and he gets $2.
25+3+2=30

No one in the scenario has any idea that there's a supposed "missing dollar."

 

[Hadoblado]

It's the power of suggestion.

Spoiler

Nobody would choose to do the math this way if they weren't influenced by the phrasing of the problem. People trust their factchecker to ring if they see a misuse of math/logic, but over-estimate this ability.

Three people check into a hotel room.

Fine.

The clerk says the bill is $30, so each guest pays $10.

Fine.

Later the clerk realizes the bill should only be $25.

Fine.

To rectify this, he gives the bellhop $5 to return to the guests.

Fine.

On the way to the room, the bellhop realizes that he cannot divide the money equally.

Fine.

As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 as a tip for himself.

Fine.

Each guest got $1 back:

Fine.

so now each guest only paid $9;

Fine.

bringing the total paid to $27.

Fine.

The bellhop has $2.

Fine.

And $27 + $2 = $29

Fine.*

so, if the guests originally handed over $30, what happened to the remaining $1?

What remaining dollar? Everything stated so far is true for the purposes of the hypothetical, but there is also no missing dollar. The quote in red is true but irrelevant, guiding the thoughts of the reader down a line of reasoning that is not appropriate. 27 + 2 = 29, but if you use labels instead of numbers you can see:

([Hotel(25)]+[bellhop(2)]=27)+[bellhop(2)]=29. Adding the bellhop twice while ignoring the money the bank accounts of the clients is not representative of the total money in this scenario, and equating it with such is wrong. The reason people do this is because if all of the explicitly stated facts are true (which they are), they assume there's nothing wrong with the problem - when the form of the suggested solution is at fault.

The correct account is [bellhop(2)]+[hotel(25)]+([Client(1)]*3), which gives the appropriate sum.

 

[Teax]

The problem starts here:

And $27 + $2 = $29

The "And" makes it sound like the equation has meaning, when in fact a "-" was replaced by a "+". Here's how it should've sounded:

Each guest got $1 back: so now each guest only paid $9; bringing the total paid to $27. The bellhop has $2. And $27 - $2 = $25 so, the hotel got the $25.

 

[Aerl]

It should be $27-$2= $25
I'm too lazy to explain it in detail.

You get $27 total payed + $3 returned.
$2 are payed to bellhop.

I just hope you follow my train of thought and this makes enough sense to you.
No riddle, just bad math.

 

[SpaceYeti]

This "riddle" is old and easy, since it's not really a riddle so much as bad math. try this one:

There are four men, each with a black or white hat on. There are only two hats of each color. Three men are lined up in a room. The men cannot see their own hat, and they're all facing the same direction. They can see only the one in front of them. That is, the first man sees nothing relevant, the second man sees only the first and his hat, and the third man sees the second and first men, whose hats are different colors. There are no mirrors or reflections, no turning their heads, they didn't see each-other before this, and no similar tricks. They only know the color of the hat on the other men's head if they can currently see it. They will be killed if they say anything except the color of the hat on their head, after which they're free to go. They live in an evil empire and the emperor is just toying with them, I guess. The fourth man, wearing the fourth hat, is in an entirely separate room, who cannot see or be seen by the first three men.

They all go free. Who deduced the color of the hat on his head, and how did he deduce it?

 

[The Grey Man]

Spoiler

The second man sees that the first man is wearing a [COLOUR ONE] hat. He deduces that, since the third man has not declared the colour of his hat to be [COLOUR TWO] (based on both hats in front of him being [COLOUR ONE], eliminating all but the [COLOUR TWO] hats), therefore his own hat must be [COLOUR TWO].

 

[SpaceYeti]

Correct.

 

[Hadoblado]

Spaceyeti do you need to get all four, or just any one?

Spoiler

I can get the first and second men to deduce their colours, but I can't seem to get further than that. Once first and second have declared their differing colours, man three and four are both entirely unobserved, thus barring the inference of what an observer would do if they could see your colour.

 

[Reluctantly]

Yeah, that's how I reasoned it. 3 dollar refund makes the total paid 27 and not 30. It only needs to add up to 27.

 

[SpaceYeti]

Spaceyeti do you need to get all four, or just any one?

Spoiler

I can get the first and second men to deduce their colours, but I can't seem to get further than that. Once first and second have declared their differing colours, man three and four are both entirely unobserved, thus barring the inference of what an observer would do if they could see your colour.

Only one. Also, please explain how the first man could possibly know the color of his hat?

 

[Aerl]

Spoiler

1st cant see any hats
2nd can see only 1st's hat
3rd can see 1st's and 2nd's hats

If 3rd can't determin his hat's color that means he sees two hat's with diferent colors, thus 2nd can tell his hat's color is opposite of 1st's.

2nd tells his hat color and walks out. Knowing this, 1st's now knows that his color of the hat is opposite of 2nd's because 3rd couldn't determin his hat's color. 1st walks out.

We're left with 3rd who knows that only one black and one white hat is left and 4th who's isolated.

 

[Hadoblado]

Oh okay.

Spoiler

Man C doesn't say anything, meaning he does not have the required information to eliminate a colour. Man B does not hear a reply from man C, and reasons that the only time man C wouldn't be able to deduce the colour of the hats in front of him is if man B and man A had different colour hats. He can see man A's hat, therefore he knows his own is of the other colour.

Man A hears man B's response, and also reasons out that man C couldn't make a judgement because A and B were different. Since man B just called out what colour his hat was and left, man A can safely called out the opposing colour and leave too.

But that was in the scenario they all individually had to figure out their colours.

 

[Reluctantly]

Oh okay.

Spoiler

Man C doesn't say anything, meaning he does not have the required information to eliminate a colour. Man B does not hear a reply from man C, and reasons that the only time man C wouldn't be able to deduce the colour of the hats in front of him is if man B and man A had different colour hats. He can see man A's hat, therefore he knows his own is of the other colour.

Man A hears man B's response, and also reasons out that man C couldn't make a judgement because A and B were different. Since man B just called out what colour his hat was and left, man A can safely called out the opposing colour and leave too.

But that was in the scenario they all individually had to figure out their colours.

Hmm, this riddle is strange then. It would suck to have man 3 be suicidal or homicidal+suicidal, since he seems to hold most of the cards, especially when man 1 and 2 have the same color hat.

 

[Pyropyro]

Guys, good math skills are awesome and all but the bellhop has not yet delivered my luggage to my room.

 

[Reluctantly]

Unfortunately, if you didn't buy the insurance package, there's nothing we can do to recover your loses. Better hope it gets delivered soon.