I thought I'd posted up this puzzle ages ago, but a Google search shows no sign of it. As I've recently been discussing it online with Chris Garratty, I thought I'd throw it open to the vast Kree Intelligence that is gamebook fandom. You've played Sorcery, right? You must all have IQs as big as the Death Star. So here goes:
While walking upon a path through unmapped territories, you come across a group of three cowled figures standing where two roads meet. You are informed by one whose counsel you have no reason to doubt that these three are Mung, who keeps the secrets of the grave (for he is the god of death) by invariably lying, Sish, the Destroyer of Hours, who speeds the flight of time's arrow by always telling the truth, and Kib, the god of life, who created Man and consequently lies and tells the truth equally without conscience. Further, you are told that one of the two roads leads to Paradise while the other takes travellers to the lowest circle of Hell. Presupposing that you wish to take the road to Paradise, how can you, by asking one yes-or-no question of one of the three gods (who are, incidentally, indistinguishable), find whether to go left or right?So, with two questions you might start with the old, "If I asked if the way to Paradise is left, would you say yes?" If you only have a liar and a truth-teller, if the answer is yes you should go left, and if it's no you go right. The snag is, that question sets up a logical impossibility for the god who randomly chooses to lie or tell the truth. Figure that he works out the lying or truthful response to the current question and then flips a mental coin to choose which to say. Your hypothetical question nested within the first is a separate question for which he hasn't yet flipped that coin.
However, if you did have two questions, there is a way to use the first question to weed out Kib. (This is from Ivan Morris, by the way, who originally devised the puzzle.) Say you pick one god (call him A) and you ask, "Is B more likely to tell the truth than C?" If you get a yes, C cannot be Kib. If you get a no, B cannot be Kib. You can then proceed to the question above.
But here's the snag. You don't have two questions, you just have one...
When you asked this on Twitter, I misunderstood and thought there was no answer, because I thought you were saying that the god who answered randomly did so by saying yes or no randomly. However, I see now that in fact the god who answers randomly does so by telling the truth or lying randomly. So, I think the solution is the following; pointing to one of the paths, you ask:
ReplyDelete"If I ask you whether that is the road to Heaven, will you answer yes?"
If it is the road to Heaven, then a god who is telling the truth will say yes. A god who is lying will also say yes, because if you simply asked them whether that was the road to heaven, they would say no.
Conversely, if it is the road to Hell, the truthful god will say no, and the lying god will also say no.
You're on the right lines, Graham. It is crucial that Kib lies or tells the truth randomly, he doesn't just say yes or no randomly.
ReplyDeleteBut... in your question, Kib is being asked what he will say on his next answer, for which he hasn't yet flipped that imaginary coin.
Suppose you were to formulate the question thus: "If I ask you whether that is the road to Heaven, will you *certainly* answer yes?" Mung and Sish both say yes (assuming it is the correct road) but Kib must think: "Well, I wouldn't certainly say yes. I might decide to lie about it. So the true answer to the question asked is no, I wouldn't certainly say yes. But if I decide to lie right now, I'll answer yes." And then he flips that coin and gives you a yes or a no - and zero information.
You are close. You just need to find a way to ask the question without making it hinge on his response to a future question. Or is that too broad a hint? ;)
If only you had the skill CUNNING, then you could throw rocks at them and get them to fight or something. With the skill FOLKLORE, you would know that Mung loves beans and therefore you can detect him by the smell. Are we allowed to summon a Faltyn? Or use the Ring of Obedient Parts and scout ahead?
ReplyDelete"If I ask you whether that is the road to Heaven, will the god on your right hand (or the other end) *certainly* answer yes?"
That probably doesn't help either. If only I'd paid attention in Sage school...
Summon a faltyn?! James, it's not that desperate!
DeleteNo Sage XPs for you, sorry.
OK, here's the solution. Look away now if you still want to have a go at figuring it out...
ReplyDeleteThe question is: "If I had asked if the way to Paradise is left, and received the answer yes, would that reply have the same degree of truthfulness as your reply to this question?"
Assuming Paradise is left, whether the respondent is in truthful or lying mode he must say yes. Note that the way the question is posed doesn't rely on Kib predicting his own reply to a future question (which he has yet to toss that mental coin for) but instead presents him with a fait accompli.
Well, I felt sure this had to involve asking a god about his own answer to a question rather than asking him about answers that the other gods would give, but even if I'd spent all day trying to figure it out, I'm not sure of the chances that a random flash of inspiration would have brought me that solution.
DeleteI suspect there's no clear method for finding a solution. A bolt of inspiration is needed. That's how I solve most problems, though.
DeleteI just want to point out that this solution is valid only when assuming the principle of bivalence. If it is possible for a reply to be both true and false at the same time, Kib may still reply arbitrarily.
DeleteThanks for sharing this puzzle!
That's why I said that Kib cannot reply to questions whose answers depend on some future, as-yet-undetermined toss of that mental coin. It's not that he then responds arbitrarily; he can't respond. This gives you a chance of identifying him, but then your question is used up.
DeleteParadise is left or right. You can't get more bivalent than that. And if you look at my formulation of the question you'll see that it avoids straying off into indeterminate speculative futures where replies risk losing that bivalence.
Ah no, I was not talking about that kind of bivalence. What I meant is that the reason Kib cannot answer arbitrarily is because one of the replies cause a logical contradiction by forcing his answer to be true and false simultaneously, which is impossible. However, that impossibility only occurs when using a bivalent logic such as Boolean algebra.
DeleteWhen using a four-valued logic having ∅, {true}, {false} and {true, false} as possible values, the contradictory answer can be assigned the value {true, false}, which is equally true and false and therefore can be said by Kib without even tossing his mental coin. I believe a good example of such a logic is Dunn/Belnap's 4-valued relevance logic system.
Who knows what kind of logic gods use?
That's a lot of theory, but I'm a practical fellow. You phrase a question, I'll tell you how they'd each reply to it.
DeleteUpdate: I had been thinking, okay, so maybe heaven and hell are quantum states, so perhaps we do need a form of logic that superposes truth and falsity. But then this week a new model has been proposed that does away with the spookiness of quantum theory and reduces the universe back to a neatly Newtonian and Aristotelian form - albeit you need an infinity of them. (Google this week's New Scientist: "Ghost universes kill Schrodinger's cat".)
DeleteIf I asked any of you "would Kib answer yes truthfully that the left hand path leads to paradise", would the answer be yes?
ReplyDeleteDoesn't work, I'm afraid. First consider the inner hypothetical question, "Would Kib answer yes truthfully that the left-hand path leads to paradise?" If paradise is left, a truthful response to that is yes and a lie is no. Then you ask if they would say yes. Sish truthfully says yes. Mung lies and also says yes. But Kib is being asked yet another of those questions that he hasn't tossed the mental coin for. He knows that he might end up saying no to the hypothetical question, but in answering your question about the hypothetical question he might truthfully say no - that is, he is truthfully telling you that he can't guarantee what his answer to a question in the future will be. That's why the solution has to be phrased in a way that doesn't ask him to predict his future hypothetical response.
ReplyDeleteIn my opinion, Graham Hart's logic is sound, because from the way things are going, you are more hung up on Kib's ambiguity than finding out the road to Paradise. Graham's question disregards Kib altogether and forces the gods who provide absolute answers to point out the road to Paradise. I may be wrong about the actual phrasing of the question, but as you mentioned, with two questions you would be able to deduce which of the three is Kib, but I believe his idea to disregard Kib altogether provides a simpler solution when forced with a "one question only" rule.
ReplyDeleteThe trouble with Graham's question is that it won't give him any information. If he gets a yes and is lucky enough to have picked Mung or Kish, he should go left. But if he picked Kib, that yes is Kib choosing to lie about the certainty of his hypothetical answer - in which case it's equally likely that paradise is to the right. So Graham's question gets him to paradise 5 times out of 6.
DeleteRight, good point. And I seem to have glossed over the part: "asking one yes-or-no question of one of the three gods". I was under the impression that the question was presented to all three of them.
DeleteMy quandary is that if the Gods ALWAYS answer truthfully (or not) does that not also mean that they ALWAYS know the truth as well? e.g. They can only answer yes or no and they must always tell the truth or lie...so I could ask will I live to be 80 years old? As they can't say "I don't know" or refuse to answer that means that they can see the future...
ReplyDeleteI think a more serious problem there is that you'd be no nearer to knowing which road to take. But okay, just assume they are not gods, they're ordinary mortals and if you ask them something they couldn't possibly know then they will just look blank and you wasted your question. Note that this can be used to identify Kib, but you then don't have a question...
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