Wizard
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August 14th, 2010 at 3:16:03 PM permalink
Three guests 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 for himself.

Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $27. The bellhop has $2. If the guests originally handed over $30, what happened to the remaining $1?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Nareed
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August 14th, 2010 at 3:25:59 PM permalink
If I say instead, "Each guest has $1 and the bellhop has $2, which totals $5," then there's no missing buck.
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Wizard
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August 14th, 2010 at 3:33:20 PM permalink
Quote: Nareed

If I say instead, "Each guest has $1 and the bellhop has $2, which totals $5," then there's no missing buck.



Yeeeeeeeeees. But much like the two envelope problem, where is the FLAW in the logic of my solution?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
mkl654321
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August 14th, 2010 at 4:52:33 PM permalink
Quote: Wizard

Yeeeeeeeeees. But much like the two envelope problem, where is the FLAW in the logic of my solution?



The flaw is in asking the question at all, as in it being conceivable that somehow in these simple transactions, a dollar could simply have disappeared into thin air. Thus do I cut the Gordian Knot of mathematics with the sword of logic.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
mkl654321
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August 14th, 2010 at 4:56:55 PM permalink
Quote: Wizard

Yeeeeeeeeees. But much like the two envelope problem, where is the FLAW in the logic of my solution?



The flaw is in adding numbers that have nothing to do with one another, differences, remainders, whatever.

Each guest owed $8.33, overpaid by $1.66, and received back $1 of the overpayment. The remaining 66 cents each (rounding up, $2) was purloined by the bellboy. Simplissimo.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
DorothyGale
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August 14th, 2010 at 5:07:22 PM permalink
Mr. W,

Instead of pointing out the flaw ... which is a big banana ... I'll just say that this is Enron accounting ... mixing up Equity Transfers with Accounts Receivable ... just turn this accounting method into a money making machine and who knows, in a few years maybe you'll be testifying before Congress ...

It is so boring here ... so flat ... so damn flat ... is there a 3-rd dimension? I mean really, is there?

--Dorothy
"Who would have thought a good little girl like you could destroy my beautiful wickedness!"
Wizard
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August 14th, 2010 at 6:35:38 PM permalink
Right, the flaw is in mixing money paid and money received. The question was, "Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $27. The bellhop has $2. If the guests originally handed over $30, what happened to the remaining $1?"

The problem is that the $27 the guests paid included the $2 the bellhop received. So that $2 is being doublecounted. Money paid should balance with money received.

Money paid = $27 ($9 from each guest)
Money received = $27 ($25 to the hotel and $2 to the bellhop)

The other $3 is money that remained with the guests.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Doc
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August 14th, 2010 at 6:46:19 PM permalink
Very well done, Wizard. I thought this problem was so old that the question and the solution were universally known by small children. Perhaps not. Anyway, that is why I made reference to the slick ruse of the problem statement in one of my posts is the two-envelope thread. Hope whatever I said there didn't spoil anyone's entertainment here.
Nareed
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August 14th, 2010 at 6:57:58 PM permalink
Quote: Doc

Very well done, Wizard. I thought this problem was so old that the question and the solution were universally known by small children.



Oh, you're thinking of this:

You face a choice between two doors, labeled A and B. One leads to freedom, the other leads to a dungeon. Naturally you don't know which is which. Each door has a guard. One guard always lies, the other always tells the truth. Again you don't know which is which. You are allowed only one question to either guard (one question total). What do you ask?


Here's one possible solution
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NicksGamingStuff
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August 14th, 2010 at 7:08:06 PM permalink
I say you ask either guard which door is not to freedom, the lying one will tell you which one is to freedom and the truthful guard will tell you the truth!
Doc
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August 14th, 2010 at 7:09:39 PM permalink
Quote: Nareed

Here's one possible solution

I like it! Nice lateral thinking. Ed de Bono would be pleased.
mkl654321
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August 14th, 2010 at 7:19:28 PM permalink
Quote: Nareed

Oh, you're thinking of this:

You face a choice between two doors, labeled A and B. One leads to freedom, the other leads to a dungeon. Naturally you don't know which is which. Each door has a guard. One guard always lies, the other always tells the truth. Again you don't know which is which. You are allowed only one question to either guard (one question total). What do you ask?


Here's one possible solution



Isn't the "classic" solution to ask either guard (it doesn't matter), pointing to either door (it doesn't matter), "if I were to ask you if this door leads to freedom (or, the dungeon), what would you say?" The catch, of course, is that either guard would answer the same way, and that answer would be the truth.

So if the dealer peeks under her hole (ten) card, and then you show her your hard 16, and you ask her, "if I were to hit this, would that be a mistake?" and since dealers always lie...
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
Nareed
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August 14th, 2010 at 7:28:07 PM permalink
Quote: Doc

I like it! Nice lateral thinking. Ed de Bono would be pleased.



I did take Rubik's Cube apart.
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Doc
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August 14th, 2010 at 8:45:20 PM permalink
Quote: mkl654321

Isn't the "classic" solution to ask ....

Of course. That's why the solution in Nareed's link is so good. It required a new level of creative thinking, and it didn't involve asking any questions at all.

I really enjoyed de Bono's books (except that after eight or ten, they started to seem redundant). Years ago I taught a class in solving open-ended problems. I required each student to select and read at least one of de Bono's books -- didn't really matter which one, except the puzzle books didn't count.
Mosca
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August 15th, 2010 at 1:45:21 AM permalink
I would ask 1 of the guards if he were a tree frog.... and does his door lead to a dungeon.
A falling knife has no handle.
Doc
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August 15th, 2010 at 4:54:10 AM permalink
Quote: Mosca

I would ask 1 of the guards if he were a tree frog.... and does his door lead to a dungeon.

Two questions. Violates the rules.
cclub79
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August 15th, 2010 at 5:42:04 AM permalink
Quote: NicksGamingStuff

I say you ask either guard which door is not to freedom, the lying one will tell you which one is to freedom and the truthful guard will tell you the truth!




That won't work. If A is freedom and B is death, and you ask the truthful one, he'll point to B. If you ask the liar, he'll point to A. Not knowing who's who, you have no new information.


Quote: mkl654321


Isn't the "classic" solution to ask either guard (it doesn't matter), pointing to either door (it doesn't matter), "if I were to ask you if this door leads to freedom (or, the dungeon), what would you say?" The catch, of course, is that either guard would answer the same way, and that answer would be the truth.



Almost, but not quite. If you ask the liar that question, and are pointing at the freedom door, he'd say "I'd say that that door leads to freedom." With your logic, you'd walk through that door into the dungeon.

The correct answer I learned was you have to say, "If I were to ask HIM (pointing at the other guard) which door is freedom, what would he say?" Again, if A is freedom and B is death, the liar will tell you that the truthful one would say B (because he's lying), and the truthful one would tell you that the liar would say B (because he's telling the truth about the liar). So you ask that question and you go through whichever door they don't say.
Mosca
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August 15th, 2010 at 7:48:58 AM permalink
Quote: Doc

Two questions. Violates the rules.



I know. It's a movie reference. The Enigma of Kaspar Hauser (Every Man for Himself and God Against Them All)

Kaspar Hauser: logic scene (4 minutes or so)

I know it's not exactly the same question, but every time I hear any variation of it I think of this. I can't help it.

"Understanding is secondary; the reasoning's the thing!"
A falling knife has no handle.
Garnabby
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August 15th, 2010 at 8:18:43 AM permalink
Quote: Nareed

I did take Rubik's Cube apart.




You're kidding, right?
Why bet at all, if you can be sure? Anyway, what constitutes a "good bet"? - The best slots-game in town; a sucker's edge; or some gray-area blackjack-stunts? (P.S. God doesn't even have to exist to be God.)
Nareed
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August 15th, 2010 at 9:03:53 AM permalink
Quote: Garnabby

You're kidding, right?



I'm kidding, wrong.

Seriously, didn't everyone take a Rubik's cube apart at some time?
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weaselman
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August 15th, 2010 at 9:46:27 AM permalink
Quote: cclub79


Almost, but not quite. If you ask the liar that question, and are pointing at the freedom door, he'd say "I'd say that that door leads to freedom." With your logic, you'd walk through that door into the dungeon.



Why would you go through it to dungeon if it leads to freedom? :)
The liar will lie about his answer, and you'll get the truth by double negation. The truthful guard will still tell you the truth.
Quote:


The correct answer I learned was you have to say, "If I were to ask HIM (pointing at the other guard) which door is freedom, what would he say?" Again, if A is freedom and B is death, the liar will tell you that the truthful one would say B (because he's lying), and the truthful one would tell you that the liar would say B (because he's telling the truth about the liar). So you ask that question and you go through whichever door they don't say.


Yes, that'll work too, but this extra level of indirection (asking what the OTHER guy would say) is not necessary.
"When two people always agree one of them is unnecessary"
mkl654321
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August 15th, 2010 at 2:29:21 PM permalink
Quote: cclub79


Almost, but not quite. If you ask the liar that question, and are pointing at the freedom door, he'd say "I'd say that that door leads to freedom." With your logic, you'd walk through that door into the dungeon.



Uh, no. I'd walk through that door to freedom, actually--since it was the freedom door I was pointing at when I asked the lying guard that question.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
Wizard
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August 15th, 2010 at 4:37:26 PM permalink
By the way, I would point to either path and ask either villager "Are you from that village?" If he said "yes," then it is the truthful village, else the lying village. Can anyone formulate a working question to the puzzle, in proper English, with fewer words?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
cclub79
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August 15th, 2010 at 5:40:15 PM permalink
Quote: mkl654321

Uh, no. I'd walk through that door to freedom, actually--since it was the freedom door I was pointing at when I asked the lying guard that question.



My apologies, I didn't follow the explanation. That absolutely works too. In high school we always had the answer of asking what the other guy would say.
mkl654321
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August 15th, 2010 at 9:05:45 PM permalink
Quote: weaselman

Yes, that'll work too, but this extra level of indirection (asking what the OTHER guy would say) is not necessary.



Strictly speaking, that won't even work at all, since (because it wasn't stipulated) it isn't necessarily so that the guards know about each others' truth-telling or lying proclivities--only that YOU do.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
weaselman
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August 16th, 2010 at 6:34:38 AM permalink
Quote: mkl654321

Strictly speaking, that won't even work at all, since (because it wasn't stipulated) it isn't necessarily so that the guards know about each others' truth-telling or lying proclivities--only that YOU do.



Good point. Yeah, that makes sense, it's better to ignore the second guard completely.
"When two people always agree one of them is unnecessary"
weaselman
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August 16th, 2010 at 6:35:53 AM permalink
Quote: Wizard

By the way, I would point to either path and ask either villager "Are you from that village?" If he said "yes," then it is the truthful village, else the lying village. Can anyone formulate a working question to the puzzle, in proper English, with fewer words?

"Is that your village?" is one word shorter :)
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Wizard
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August 16th, 2010 at 10:27:53 AM permalink
Quote: weaselman

"Is that your village?" is one word shorter :)



Nice. We're down to four words. Three anybody?
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Mosca
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August 16th, 2010 at 10:50:43 AM permalink
Quote: Wizard

Nice. We're down to four words. Three anybody?



You from there? (Use inflection to convey the interrogative "are".)
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mkl654321
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August 16th, 2010 at 11:50:11 AM permalink
Quote: Wizard

Nice. We're down to four words. Three anybody?



Sure. I can do better than that:

Grab the guy by the scruff of his neck, drag him to one village or the other, and start beating him up. If someone tries to stop you, that's probably his village. If the villagers gather around and start cheering and/or placing bets, he's from the other village.

Zero words.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
Ayecarumba
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August 16th, 2010 at 12:03:44 PM permalink
Quote: Wizard

Nice. We're down to four words. Three anybody?



(Pointing) Your Village?
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