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# Hotel Bill

62 replies to this topic

### #11 redspider

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Posted 01 August 2007 - 03:54 AM

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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### #12 mdsl

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Posted 01 August 2007 - 01:07 PM

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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### #13 showman

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Posted 06 August 2007 - 12:56 AM

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...._Descending.jpg
- Escher's Cube: http://www.cs.techni...beRealFront.gif
- Waterfall: http://en.wikipedia....r_Waterfall.jpg
- Escher's Relativity: http://en.wikipedia...._Relativity.jpg
- Drawing Hands: http://en.wikipedia....rawingHands.jpg

Music
- Listen to this one to get an idea what I am talking about:
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

- http://en.wikipedia....ki/Strange_loop
- Douglas Hofstadter - Gödel, Escher, Bach (http://en.wikipedia....l,_Escher,_Bach)
- Laszlo Mero - Habits of Mind: The Power and Limits of Rational Thought (http://en.wikipedia....iki/László_Mérő)
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### #14 bonanova

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Posted 23 August 2007 - 10:14 AM

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?
Spoiler for ...

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Vidi vici veni.

### #15 colddog

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Posted 12 September 2007 - 08:24 PM

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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### #16 ectolo

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Posted 19 September 2007 - 11:03 PM

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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### #17 BoilingOil

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Posted 26 September 2007 - 02:03 AM

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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### #18 Shauna

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Posted 02 October 2007 - 03:45 PM

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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### #19 __amyyyy

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Posted 06 October 2007 - 02:05 AM

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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### #20 slmo

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Posted 10 October 2007 - 06:15 PM

[quote="bonanova":3054b]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?
Spoiler for ...

ynnuf t'nsi aixelsyD
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