Another Oldie but Goodie -- The Missing Dollar Problem
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?
Quote: NareedIf 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?
Quote: WizardYeeeeeeeeees. 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.
Quote: WizardYeeeeeeeeees. 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.
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
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.
Quote: DocVery 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
Quote: NareedOh, 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...
Quote: DocI like it! Nice lateral thinking. Ed de Bono would be pleased.
I did take Rubik's Cube apart.
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.Quote: mkl654321Isn't the "classic" solution to ask ....
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.
Two questions. Violates the rules.Quote: MoscaI would ask 1 of the guards if he were a tree frog.... and does his door lead to a dungeon.
Quote: NicksGamingStuffI 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.
Quote: DocTwo 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!"
Quote: NareedI did take Rubik's Cube apart.
You're kidding, right?
Quote: GarnabbyYou're kidding, right?
I'm kidding, wrong.
Seriously, didn't everyone take a Rubik's cube apart at some time?
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.
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.
Quote: mkl654321Uh, 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.
Quote: weaselmanYes, 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.
Quote: mkl654321Strictly 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.
"Is that your village?" is one word shorter :)Quote: WizardBy 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?
Quote: weaselman"Is that your village?" is one word shorter :)
Nice. We're down to four words. Three anybody?
Quote: WizardNice. We're down to four words. Three anybody?
You from there? (Use inflection to convey the interrogative "are".)
Quote: WizardNice. 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.
Quote: WizardNice. We're down to four words. Three anybody?
(Pointing) Your Village?
