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Topic: NEW PROBLEM: The Gods of Gibberland (Read 18332 times) |
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Joshua Franklin
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #25 on: Aug 10th, 2002, 12:42pm » |
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Haven't looked at the hint yet, only been thinking about the problem for 24 hours, maybe I'll look at it before I go to work. I'm not really closer to an answer...I think I need to get smacked upside the head so I start thinking differently. 
Eric--this problem is a marvel. The XOR really turns things around...would almost be easier if GibberKnexus answered randomly.  Thanks!
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #26 on: Aug 10th, 2002, 1:00pm » |
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Thanks dude!!! I really appreciate your support. I was pretty excited when I came up with the xor, too; it really shatters the standard da/ya construction!
Sorry to hear you have to work on a Saturday. (Nice pic btw! I came close to taking that one myself!)
Best,
Eric
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icon
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #27 on: Aug 10th, 2002, 5:30pm » |
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eric what a riddle!
very nice, i spend some time on it and i didnt get anywhere far except i think the key is to have the 1st question based on the xor's answer so to speak i could be totally off but i go back to think
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #28 on: Aug 10th, 2002, 7:24pm » |
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Thanks Icon! Glad you are enjoying the riddle!! I am also honored to be graced by your first post as a true member -- that pic is totally you!
Best,
Eric
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tim
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #29 on: Aug 11th, 2002, 3:04am » |
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I am pretty sure I have a general framework for solving it, but there is substantial ambiguity in the wording of the problem. I have used the framework to draw up three sample solutions, each with different interpretations.
The central problem is that when you say "knexuses always answer the xor of what the knight and knave would answer", how exactly does this work?
As one example, suppose the gods in positions 1, 2, and 3 are GibberKnexus, GibberKnave, and GibberKnight, repectively. Now you ask GibberKnexus "Is a knight in position 1?"
1) Knexuses answer the xor of "what the knight and knave would answer". If he were a knave, he would answer yes because it would be false. If he were a knight, he would answer yes because it would be true. Hence GibberKnexus should answer no.
2) If you actually ask any knave, he would answer yes. If you actually ask any knight, he would answer no. Hence GibberKnexus should answer yes.
Should he answer yes or no?
Basically, you use a counterfactual in the definition of the problem without defining the parameters under which the counterfactual is evaluated. There are actually a lot more than just two cases. That's why I have three solutions, and they don't come close to covering all the possibilities. 
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Joshua Franklin
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #30 on: Aug 11th, 2002, 9:30am » |
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tim,
maybe i've been looking at that the wrong way, but when I come up with a question that I would ask them, I make a little chart and map out what the 3 gods would say...In your example:
Is GibberKnight in position 1 (no)?
GibberKnight says "no", GibberKnave says "yes", and therefore GibberKnexus would say "yes".
You shouldn't switch their physical positions to know what the XOR is; keep the same conditions for all 3 possible answers (+,-,XOR) and treat is more as a "what if".
I've been treating the XOR such as:
If both GibberKnight and GibberKnave would answer the question (regardless of if it is asked) in the same way (ie. Yes and Yes, or No and No), the XOR returns a False, or "no". However, if GibberKnight's and GibberKnave's answers differ (Yes and No, or No and Yes), the XOR returns a True, or "yes".
However, I still haven't come up with the answer so I could be doing it incorrectly too. 
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #31 on: Aug 11th, 2002, 9:42am » |
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Tim,
That is a great question; I wish you had asked it earlier so that I could clarifiy it for the other members of this forum -- although my own solution is independent of the answer!
Your second proposition is the correct interpretation; if it were the first case I would have written "the xor of what he would answer if he were a knight and what he would answer if he were a knave." But you are right that there are some subtleties involved.
Here the definition is meant as the simplest possible interpretation, i.e., you get the xor of how each of a knave and knight would respond to the question. All "hard" references still refer lexically to the "true-life" situation, as in your example where "god 1" still refers to GibberKnexus. All "soft" references such as "you" in "are you a knight?" refer dynamically to the new persona. But only in the sense of identity! Everything else (e.g. position) still holds to true life, as if a virtual knight or knave shared all non-identity characteristics other than potential equivalence to hard references.
Example questions and answers with your set up:
1. Is god 1 a knight? (The example from your post.)
"Yes."
2. Are you a knight?
"No." (Though otherwise equivalent, there is a difference in scoping.)
3. Are you in position 1?
"Yes." (No change in position, which is a non-identity characteristic.)
4. Are you a knight and are you in position 1?
"No." (The virtual knight and knave would both answer "yes." Room for seeming paradoxes here
for the very very detail-oriented, but hey, it's more power to the people! And consistency is the
most important thing.)
Does that clarify all your questions? Does everyone agree that these rules are self-consistent (if sometimes seemingly paradoxical) and make intuitive sense? Again, I do not believe it affects the solvability of the problem, so I am writing here what my intuition was when I first wrote the problem.
(If the seeming paradoxes bother people, feel free to assume either fully lexical or fully dynamic scoping -- if I'm not missing something, they just make the problem harder anyway!)
Finally Tim, I'd love to see your solns -- off-line so as not to spoil it for the others, if you wouldn't mind. If that would be ok, please feel free to e-mail me (eyeh@post.harvard.edu).
Best,
Eric
PS Just read Josh's post, which sounds about right without going into some of the intricasies.
PPS Damn I can't stand this editor!!! The edit window, preview window, and true post window all have different horizontal buffers, so you can't figure out how to align split lines at all without officially posting!!! It's taken me several attempts just to get my question indents to line up, mostly because I first trusted the editor and then the preview! Any ideas?
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| « Last Edit: Aug 11th, 2002, 9:52am by Eric Yeh » |
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Jonathan_the_Red
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #32 on: Aug 14th, 2002, 2:22pm » |
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I haven't been able to dedicate many brain cycles to this problem lately (I do have a life outside of puzzles, unfortunately) but I'm starting to wonder if it's possible.
Potential proof of impossibility:
The first question gives you no immediate information, since you don't know what the word means. The second answer has two possibilities: same as the first, or different. Of the six possibilities in the solution set, the best you can do after two questions is divide them into two sets of three. (not that it's easy to do even that much.) There's no way that the third question, with only two possible answers, can produce a unique solution out of three possibilities.
Where am I going wrong here, Eric?
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #33 on: Aug 14th, 2002, 2:38pm » |
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Jon,
First of all, it's good to have a life outside of puzzles!!!  So don't worry, I'm not sitting here ticking away the clock so that when you post an answer I can finally declare that "it took Jon a whole ten days to do it"!
Second, if it's any consolation, Tim has also found a solution, and also agrees with mine. So you can rest assured it's solveable.
Finally, where are you going wrong?
You can extract more information out of the first question, even if not immediately.
Let me know if you need more.
Best,
Eric
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Rupert
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While I haven't been able to solve it yet, how about a proof that it is unsolvable?
There are two kinds of answers:
1. Those which all gods would answer the same:
They give you no clue about who is who, but they solve the language problem. That leaves you with two yes/no answers which can produce four different results. Not enough for the six possible combinations of gods.
2. Those which not all gods would answer the same:
Having no knowledge of the language, the answers would be
one of the four combinations:
1. foo foo foo
2. foo foo bar
3. foo bar foo
4. foo bar bar
(assuming that you call the first answer you hear 'foo', the other one 'bar')
So, again, just four possible answers for six combinations.
Duh.
While this one would be pretty without the language issue, I don't see how it could be possible to solve. Did I misread something?
BTW: I still don't see how the language problem in PPF was solved (it was still hard enough without, though  ): Jon's solution "Does 'splunge' mean 'yes' if and only if P?" doesn't work, unless you know beforehand that splunge means yes: If splunge doesn't mean yes, the answer will always be no. And, to be picky: If splunge does mean yes, the answer would still always have to be no, because splunge does not only mean yes when P, but it always means yes.
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #35 on: Aug 14th, 2002, 3:14pm » |
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Rupert,
No, you did not misread anything, but you are missing a critical innovation for why your second "four combination" argument doesn't hold. Don't worry though, you're in good company -- only two people in the world have gotten it so far!!!
Re: The PPF language problem: Note that the "iff" operator returns T on (F,F).
Best,
Eric
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Jonathan_the_Red
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Re: Language Barrier
« Reply #36 on: Aug 14th, 2002, 3:16pm » |
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on Aug 14th, 2002, 3:06pm, Rupert wrote:BTW: I still don't see how the language problem in PPF was solved (it was still hard enough without, though ): Jon's solution "Does 'splunge' mean 'yes' if and only if P?" doesn't work, unless you know beforehand that splunge means yes: If splunge doesn't mean yes, the answer will always be no. And, to be picky: If splunge does mean yes, the answer would still always have to be no, because splunge does not only mean yes when P, but it always means yes.
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The truth table for "P if and only if Q" looks like this:
P Q P iff Q
T T T
T F F
F T F
F F T
If you want, you can think of it as "P is equivalent to Q." Or "(P if Q) and (Q if P)." Or "P XNOR Q."
Now, note that when P is true, the value of P iff Q is Q. When P is false, the value of P iff Q is (not Q). Let 'P' be the proposition that 'splunge' means yes. If P is true (i.e. splunge means yes), the correct answer to P iff Q is Q, meaning 'splunge' if Q is true, 'fleen' if Q is false. If P is false ('fleen' means yes, 'splunge' means no), the correct answer to P iff Q is (not Q), meaning 'fleen' (yes) if Q is false, 'splunge' if Q is true. In either case, 'splunge' is the answer if Q is true, 'fleen' if Q is false.
Sorry if this isn't very clear
Oh, BTW, Eric: the "Red" in my name is a hair color thing
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #37 on: Aug 14th, 2002, 3:26pm » |
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I like my explanation better. I'm terse.
Gee Jon, it sure took you long enough to answer that one!! And from far away across threads, too!!! Haha jk, tx for the response.
Best,
Eric
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Rupert
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Jon and Eric, thanks for explanation.
But that still requires you to know beforehand that 'splunge' is one of the possible answers, doesn't it?
But it also means that this riddle here might be solvable...
Must think...
it might...
wait...
I almost had it! 
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #39 on: Aug 14th, 2002, 3:42pm » |
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That's right -- you do need to know that it is one of the possible answers in PPF.
'Gibberland' is different.
If you get the answer, don't post it right away! E-mail me!
Best,
Eric
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Jonathan_the_Red
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #40 on: Aug 14th, 2002, 5:11pm » |
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Okay, so let's do a bit more thinking out loud...
Above, I argued that after the first two questions, you couldn't distinguish between getting AF and FA, nor could you distinguish between getting AA and FF. Apparently, that's incorrect
Setting aside for the moment the question of why that's incorrect, that still leaves four 'buckets' to sort the 12 possibilities into. No way to deal them out such that we can pinpoint the actual possibility with one more question. But there's an escape clause: we don't need to determine exactly which of the 12 is correct, because we can solve the problem without figuring out what word means 'yes' and what word means 'no'.
But in order for this to help us, we still need each bucket to have no more than two distinct possibilities... that is, two distinct arrangements of the three gods. Since some buckets must have more than two possibilities, that means that we need some questions that, given the same arrangement of Gods, would be answered the same regardless of which word means 'yes'.
Apologies if I'm not being very clear... it's been a long week and it's only Wednesday Let me try it like this:
Suppose the two words are 'arglebargle' and 'foofooraw'... again, you don't know either of these words prior to asking any questions. Label the gods T, F, and X. Label the 12 possibilities for god arrangement + meaning of 'arglebargle' like so:
TFXY
TFXN
TXFY
TXFN
FTXY
FTXN
FXTY
FXTN
XTFY
XTFN
XFTY
XFTN
Now, you have four buckets labeled 'A A', 'A F', 'F A', and 'F F', representing the answers you get for the first two questions you ask. Each of the 12 possibilities must go into one and only one bucket. Each bucket can contain no more than two possibilities that differ in their first three letters.
The conclusion from the above is that at least two buckets (and possibly all four) must contain multiple possibilities that differ only by last letter. These represent questions that given gods would answer the same way no matter what 'arglebargle' means. I certainly hope there's a flaw in this reasoning, because I see no way to do this
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Jonathan_the_Red
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #41 on: Aug 14th, 2002, 5:19pm » |
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By the way, I hope this puzzle doesn't involve trickery like the below:
Ask God A: Is this sentence true if and only if God A is a Knexxus?
Neither a knight nor a knave could give any answer at all, while a Knexxus would answer Yes.
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| « Last Edit: Aug 14th, 2002, 5:20pm by Jonathan_the_Red » |
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AlexH
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #42 on: Aug 14th, 2002, 5:38pm » |
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Jonathan: Good analysis, but you're giving up too early. Consider using arglebargle or foofooraw in your questions.
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #43 on: Aug 14th, 2002, 6:03pm » |
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Jon,
I agree with the first half of your first msg. Re: the second half, your last paragraph is flawed. There is no flaw in your other logic, but there is still a soln!
Re: your second msg, no, I don't use such treachery! Note to self: add "no self-referential questions" to standard rules.
Alex,
do you have the soln?
Best,
Eric
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Rupert
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Argh -- this is driving me nuts. It's 5:20 in the morning now and I can't sleep.
Still no solution, but some more thoughts / assumptions / summary.
Please tell me if I'm already wrong here:
a) For starters, forget the whole Truth/Lie/Xor thing. Just assume they all tell the truth. Doesn't exactly make it easier, but a little less mind warping.
b) We cannot afford a whole question just to find out what yes/no means.
c) We cannot solve without ever finding out what yes/no means.
d) For questions 2 and 3, we could get yes/no answers (Using Jon's iff technique), but that still doesn't mean we know what the answer to Q1 was.
So somehow, we need questions that contain information about both: positioning and yes/no meaning. (3 questions give 8 possible outcomes. We need only six combinations. So somehow 1/4 of every answer should go towards solving yes/no. That's a 3/4 answer all in all. Whoa. I need sleep.)
One idea I had was letting the gods XOR their answer to Q2,Q3 with our assumption about whether A1 meant yes or no. But so far it's just that. An idea.
Night.
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Jonathan_the_Red
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Re: Brainstorm
« Reply #45 on: Aug 14th, 2002, 8:57pm » |
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on Aug 14th, 2002, 8:44pm, Rupert wrote:
a) For starters, forget the whole Truth/Lie/Xor thing. Just assume they all tell the truth. Doesn't exactly make it easier, but a little less mind warping.
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Well geesh, that would make it a whole lot simpler
Define the relation "reliability" such that GibberKnight is more reliable than GibberKnave is more reliable than GibberKnexus.
Q1: Is 2+2 equal to 4?
Q2: Are you more reliable than that guy?
Q3: Is that guy more reliable than that other guy?
Quote:
c) We cannot solve without ever finding out what yes/no means.
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I don't think this is the case. In fact, I'm pretty near certain that in at least some cases we will figure out which God is which without knowing what yes/no means.
Quote:
One idea I had was letting the gods XOR their answer to Q2,Q3 with our assumption about whether A1 meant yes or no. But so far it's just that. An idea.
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Interesting idea. I'll mull it.
While I'm at it, I want to run something by Eric:
(hidden)
I'm exploring possibilities for the first question that involve things like "P if and only if the word for 'yes' is alphabetically before the word for 'no'" I don't want any hints on this yet, just wanted to say I'm exploring it
(end of hidden)
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Night.
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Get some sleep
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| « Last Edit: Aug 14th, 2002, 8:58pm by Jonathan_the_Red » |
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #46 on: Aug 14th, 2002, 8:58pm » |
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Rupert,
Sorry to cause you so much trouble!!! Where are you, then, London?
a) A good primer for the first innovation, I agree.
b) True.
c) False.
d) True and True.
Best,
Eric
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Eric Yeh
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Re: NEW PROBLEM: The Gods of Gibberland
« Reply #47 on: Aug 14th, 2002, 9:02pm » |
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Erg, you always beat me to the post!!
Ok then Jon, thanks for the update. No hints from me. Enjoy the exploration!
Best,
Eric
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Jonathan_the_Red
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Re: Brainstorm
« Reply #48 on: Aug 14th, 2002, 9:11pm » |
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Quote:
d) For questions 2 and 3, we could get yes/no answers (Using Jon's iff technique), but that still doesn't mean we know what the answer to Q1 was.
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BTW, that bastard GibberKnexus completely blows away the iff technique. Take a simple proposition P and ask a Knight or a Knave "does plonk mean 'yes' if and only if P", and the Knight will answer plonk if P is true and blatz otherwise, while the Knave will answer blatz if P is true and plonk otherwise. The Knexus, though, will answer whichever word means yes. No longer does the iff technique break through the language barrier.
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Rupert
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on Aug 14th, 2002, 8:57pm, Jonathan_the_Red wrote:
Define the relation "reliability" such that GibberKnight is more reliable than GibberKnave is more reliable than GibberKnexus.
Q1: Is 2+2 equal to 4?
Q2: Are you more reliable than that guy?
Q3: Is that guy more reliable than that other guy?
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If the answer to Q3 is no, you're done. But if it is yes, you still don't know if 'that other guy' is between 'you' and 'that guy' or if he is more reliable than 'you'. -- i.e. the order established by Q2 leaves three 'emtpy spots' from which we would have to decide by just one question.
I hope that I'm wrong here, because that would make it easier, but I'm afraid I' not.
Quote:
I don't think this is the case. In fact, I'm pretty near certain that in at least some cases we will figure out which God is which without knowing what yes/no means.
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I came up with many solutions that work in some cases . I meant that a
general solution would have to include finding out what yes/no means.
BTW, I dreamt of this! I was in a cafeteria with two waiters, one of them needed to be lied to, the other needed the truth...
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