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# Hotel Bill

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Hotel Bill - Back to the Logic Puzzles

Three people check into a hotel. They pay \$30 to the manager and go to their room. The manager finds out that the room rate is \$25 and gives the bellboy \$5 to return to the guests. On the way to the room the bellboy reasons that \$5 would be difficult to split among three people so he pockets \$2 and gives \$1 to each person. Now each person paid \$10 and got back \$1. So they paid \$9 each, totaling \$27. The bellboy has another \$2, adding up to \$29.

Where is the remaining dollar?

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Hotel Bill - solution

This is a nice nonsense. Each guest paid \$9 because they gave \$30 and they were given back \$3. The manager got \$25 and the difference (\$2) has the bellboy. So it is nonsense to add the \$2 to the \$27, since the bellboy kept the \$2.

Three people check into a hotel. They pay \$30 to the manager and go to their room. The manager finds out that the room rate is \$25 and gives \$5 to the bellboy to return. On the way to the room the bellboy reasons that \$5 would be difficult to share among three people so he pockets \$2 and gives \$1 to each person. Now each person paid \$10 and got back \$1. So they paid \$9 each, totalling \$27. The bellboy has \$2, totalling \$29.

Where is the remaining dollar?

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• 1 month later...
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Actually, I got a totally different answer - and please tell me if I'm wrong:

For the money exchanged: The true bill was \$25 and the bellboy gave each of the three guests a \$1 back. Now, the total exchanged is \$28. And the bellboy has \$2. That equals the \$30.

For the intangible part: You can't figure in that each guest paid exactly \$9 for the room because the room bill was \$25 which can't be divided between 3 people evenly. Each person would have paid exactly \$8.33333333333333 each.

Take the \$8 x's 3 people = \$24

Plus \$3 they were given back = \$3

.33333333333333 x's 3 people = \$1 <--- the missing dollar

and the bellboy has \$2

Total = \$30

- Hope I did this right

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This has always been one of my favorite logic puzzle due to its nature of trying to misinform you and make you run with that information. The first time I heard it I spent hours trying to figure out what was up with the 29 dollars before realizing it had nothing to do with the problem, each person paid 9 dollars because the money the bell boy took is factored into their cost.

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• 4 weeks later...

great example of why math can't be done under the wrong assumptions.

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• 2 weeks later...

each person didn't pay \$9 because the total bill is \$25. \$2 was subtacted out of the equation unlike everything else was divided and multiplied. you can't multiply and divided and add and subtract in any order you want, do to order of operations. (PEMDAS)Perentases, exponents, multiply, divide, add, subtract. so remember to Please Excuse My Dear Aunt Sally.

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My logic:

Since three dollars were returned, you're trying to get \$27 in total rather than \$30. And since they each got a dollar back, they paid \$9 each to get \$27 (\$25 to the manager and \$2 to the bellboy).

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They got \$3 back. So they paid \$27, inclusive of the hotel bill(\$25) and the tip to the bellboy(\$2). So if you add \$2 to \$27, it is doing it twice.

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again proof that math calculated incorrectly is always incorrect. exception to rule: I spend more than I make! GENIUS!

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The variation of this goes an older couple chk into same hotel. They get charged same \$30. The fee is \$25. Bellboy is given same \$5.00 and this time decides to keep \$3, giving them back \$1 each. Now they paid \$14 each for the room, plus the \$3 in the bellhop's pocket. It equals \$31. ****There is the other dollar*****

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That guy must be pretty cheap not to pay for his wife's half of the room AND motivate the bellhop to tip himself.

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• 1 month later...

try not considering each person seperate but as 1 single party. 30 were given to the manager, he sent back 5 dollars and the bellhop kept 2 giving the others their 3. 25 for the room+3 returned+2 kept by the bellhop equals 30. the reason the dollar goes missing is because you give to back to each person then multiply and EXACTLY how it goes missing im not too sure. im sure someone here knows.

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EXACTLY how it goes missing im not too sure. im sure someone here knows.

think of it in steps like this.

manager + employee + customers = \$30

\$0 + \$30 + \$0 = \$30 cusomers give \$30 to employee

\$30 + \$0 + \$0 = \$30 employee gives \$30 to manager

\$25 + \$5 + \$0 = \$30 manager gives \$5 to employee

\$25 + \$2 + \$3 = \$30 employee gives \$3 to customer, keeping \$2

No dollars are lost this way.

manager + employee + customers = \$0

\$0 + \$0 + \$0 = \$0 nobody has anything

\$0 + \$30 + -\$30 = \$0 cusomers give \$30 to employee

\$30 + \$0 + -\$30 = \$0 employee gives \$30 to manager

\$25 + \$5 + -\$30 = \$0 manager gives \$5 to employee

\$25 + \$2 + -\$27 = \$0 employee gives \$3 to customer, keeping \$2

This is where the nonsense comes in. You are asked to add \$2 to the (positive) cost of the customers which gives you \$29, while if you added it to the negative cost, you would get -\$25, the opposite of the price for the room.

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Actually this teaser is one of my favorites. I've read about it first in Laszlo Mero's book "Habits of Mind: The Power and Limits of Rational Thought".

It goes much deeper into our way of thinking than it seems. What we see here is a tangled hierarchy.

The teaser achieves a nice smooth context switch, everything seems fine when one skims over the details but the outcome is just striking, an apparent paradox.

The trick is achieved by summing two values, that came from different contexts:

context I:

27 (what was actually paid) + 3 (what was given back) = 30

context II:

25 (the total price of the rooms) + 3 (what was given back) + 2 (what remained at the bellboy) = 30

And here is the tangled "hierarchy", the smooth context switch between I and II, taking the first piece from context I and the third piece from context II:

27 (what was actually paid) + 2 (what remaind at the bellboy) = 29

So we people are likely to mix contexts easily, and when the context switch is done nicely, one can experience similar odd puzzled feeling in front of the apparent paradox.

Examples of such tricks in other domains:

Painting

M.C. Escher

- Ascending and Descending - http://en.wikipedia.org/wiki/Image:Ascendi..._Descending.jpg

- Escher's Cube: http://www.cs.technion.ac.il/~gershon/Esch...beRealFront.gif

- Escher's Relativity: http://en.wikipedia.org/wiki/Image:Escher%27s_Relativity.jpg

- Drawing Hands: http://en.wikipedia.org/wiki/Image:DrawingHands.jpg

Music

- Listen to this one to get an idea what I am talking about: http://www.noah.org/science/audio_paradox/endless.mp3

When listening put the repeat on (playback should automatically restart when it reaches the end)

This is the audio equivalent of the endless staircase illusion made by Escher in the Ascending and Descending masterpiece.

J. S. Bach

- Musicalisches Opfer/Canon per Tonos

- Douglas Hofstadter - GÃ¶del, Escher, Bach (http://en.wikipedia.org/wiki/G%C3%B6del%2C_Escher%2C_Bach)

- Laszlo Mero - Habits of Mind: The Power and Limits of Rational Thought (http://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_M%C3%A9r%C5%91)

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• 3 weeks later...

Here's another variation. A man took his bride to the same hotel the next night. When he settled the bill in the morning the clerk said that will be \$9 apiece. The man dutifully paid \$81 and left, mumbling to himself. Why?

you shouldn't have to peek for this answer ...

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• 3 weeks later...

The trickery is in the wording. Say it correctly as - Three times nine is 27, minus the two kept by the bellboy equals 25 - the cost of the room.

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Think of it like this.

1 = person 1

2 = person 2

3 = person 3

b = bellboy

m = manager

1 2 3 b m

10 10 10 0 0 (total 30)

0 0 0 0 30 (total 30)

0 0 0 5 25 (total 30)

1 1 1 2 25 (total 30)

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This is a perfect example of reverse calculation.

The three men together have paid \$27, \$2 of which went to the bellboy, and \$25 of which went to the Hotel owner. The remaining three dollars were returned to them, \$1 each.

So, clearly, there is no missing dollar!

BoilingOil

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So actually the guys paid \$27 for a \$25 dollar room.

This gave the manager \$25 and the bellhop \$2. (\$3 went back to them from the 30 so the 3 is no longer involved) So they are still owed 2 bucks for overpayment on the room!!

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Is it that, they paid \$25, and if they got \$3 back, that would be \$28 and plus the \$2 the belloboy had, that would equal to \$30.

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Here's another variation. A man took his bride to the same hotel the next night. When he settled the bill in the morning the clerk said that will be \$9 apiece. The man dutifully paid \$81 and left, mumbling to himself. Why?

you shouldn't have to peek for this answer ...

ynnuf t'nsi aixelsyD

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that is freakin crazy, i spent 5 min trying to figure it out...

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lol i got confused a little in the begining but i found how simple it was after.

the bellboy has \$5 and he gave each of the three people \$1. Which would be \$5 - \$3= \$2.

so the one of the three guest has the remaining dollar.

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• 2 weeks later...

\$25 for room + \$2 for bellboy + \$3 back = \$30

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its supposed to be \$28 for the group

30 - 5 = 25 + 3 = 28 +2 = 30

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• 3 weeks later...

Three people check into a hotel. They pay \$30 to the manager and go to their room. The manager finds out that the room rate is \$25 and gives \$5 to the bellboy to return. On the way to the room the bellboy reasons that \$5 would be difficult to share among three people so he pockets \$2 and gives \$1 to each person. Now each person paid \$10 and got back \$1. So they paid \$9 each, totalling \$27. The bellboy has \$2, totalling \$29.

Where is the remaining dollar?

He takes the five dollars & gives three back to the people \$25 + \$3 = \$28 + the \$2 pocketed = the original \$30

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