# The way to Acarien, with Knights and Knaves

Once again, you're on your way to Acarien. You find yourself in a land of Knights and Knaves, who understand your language but don't speak it. You understand nor speak their language, but you do know their words for yes and no, "yok" and "pom". Unfortunately, you don't know which is which. And of course, as always, Knights always tell the truth, while knaves always lie.

You find yourself at a fork in the road, unsure which road to take, when you see someone coming down one of the roads you are considering. You don't know if it's a knight or a knave.

Is there a question you can ask, one single question, answerable with "yok" or "pom", that tells you which way to go?

I do not know if such a question exists, so answers can be either a question that works, or proof that no such question exists.

Let's test the "If I were to ask you "is left is the correct path?" would you say "yok"?" question.

$$\begin{array}{ccc|c} \text{opponent} & \text{yok} & \text{Acarien} & \text{answer} \\ \hline \text{knave} & \text{yes} & \text{left} & \text{yok} \\ \text{knight} & \text{yes} & \text{left} & \text{yok} \\ \text{knave} & \text{no} & \text{left} & \text{yok} \\ \text{knight} & \text{no} & \text{left} & \text{yok} \\ \hline \text{knave} & \text{yes} & \text{right} & \text{pom} \\ \text{knight} & \text{yes} & \text{right} & \text{pom} \\ \text{knave} & \text{no} & \text{right} & \text{pom} \\ \text{knight} & \text{no} & \text{right} & \text{pom} \end{array}$$

As shown above I've tried all possible combinations of answerer type, language, and correct path. They all seem to work well.

• I think it works... I was trying to adapt it without checking leaving it as-is first – d'alar'cop Oct 24 '14 at 15:22
• This logic is infinitely recursive in the case of the Knave, I think. The Knave must always lie. This question, therefore, catches him in a paradox; the Knave must lie about lying, which is the truth, which he must lie about. – KeithS Nov 18 '14 at 18:57
• @KeithS There is no recusion much less infinite recursion but you might mean something different by that statement. As the answer stands, pom means go right while yok means go left. If you think this doesn't work for some situation please let me know which. – kaine Nov 19 '14 at 13:50

This reduces to the Knights and knaves question.

Make an assumption about which of "yok" and "pom" translates to yes and no. If your assumption is incorrect the effect will be that from your perspective the knight will behave as a knave and vica versa.

The only refinement that you need to be careful about is that you need to ask about the word he would answer with. Ie. you need to use formulations such as "would you answer 'yok'" rather than formulations such as "would you agree".

The question is simple:

"If you were the opposite of what you are, and I were to ask you if left is the correct path, would you say yok?"

Assuming left = correct, yok = yes, person = Knight: "pom"(no)

Assuming left = correct, yok = yes, person = Knave: "pom"(no)

Assuming left = correct, yok = no, person = Knight: "pom"(yes)

Assuming left = correct, yok = no, person = Knave: "pom"(yes)

Assuming left = incorrect, yok = yes, person = Knight: "yok"(no)

Assuming left = incorrect, yok = yes, person = Knave: "yok"(no)

Assuming left = incorrect, yok = no, person = Knight: "yok"(yes)

Assuming left = incorrect, yok = no, person = Knave: "yok"(yes)

• Did you trying the simpler question? I can't find a situation where the "you were the opposite of what you are" clause is needed. – kaine Oct 24 '14 at 15:42
• 5th situation in yours:"Knave and yok=yes and left = incorrect His answer is pom" - This is incorrect – Leo Oct 24 '14 at 15:46
• the interior question's answer is "no"; he wants to say "yes"; that is "yok". The exterior questions answer is "yes"; he wants to say "no"; that is "pom". Am I missing something? I am seriously asking because it does logically seem like the questions shouldn't still work. – kaine Oct 24 '14 at 15:49
• I suppose you are technically right, particularly because your question has quotations around "yok". If you did not have quotations around "yok" and it was interpreted as a translation to the travelers language then (at least) the 5th situation would fail. Assuming, this is a spoken interaction (vs written), my question is more robust to the interpretation. – Leo Oct 24 '14 at 15:53
• KeithS formulation of counterpart in the other order is better than"if you where opposite of what you are". It's not unambigous what opposite means for persons. And also, it's of course completely unecesarry to invoke an other person at all. – Taemyr Nov 19 '14 at 10:15

"Would your counterpart in the other Order tell me to go left?" - if the answer is "yok", take the right fork, "pom", take the left.

This is a related problem to the "two doors, two men, one always lies, one always tells the truth" family of riddles. You basically need to formulate a question that will be answered the same way by either a knight or knave, thereby removing that variable from the puzzle, making the real answer clear.

The wrinkle from the original formulation is that there's only one person to ask. This isn't really a problem, because you can only ask one question in the classic form of the puzzle anyway, so in that form it merely makes the question easier to formulate. In this case, we can "invent" a second person, the opposite to the first, in our question.

By building a simple logic table for the possible variables in the puzzle (Knight or Knave, left or right), you can see that, regardless of whether the approaching traveler is a Knight or Knave, when you ask the question in my solution,

the lie incorporates the truth and vice-versa; the Knight would truthfully say that the Knave would lie, while the Knave would lie about the Knight telling the truth. The answer, therefore, is always a lie; The Knight will truthfully predict the Knave's lie, while the Knave will lie about the Knight's truthful answer. The positive answer, "yok", would at face value indicate that the left fork is correct, while the negative answer "pom" would indicate you should take the right fork, but since we know the answer is always wrong because it includes the Knave's lie, it's the opposite; "yok" means go right, "pom" means go left.

• This cannot be correct because you don't know what yok and pom mean and do not distinguish between them in any way in the question. – kaine Nov 18 '14 at 19:35