# The directions from another language?

You're out on an adventure, when you find you're lost. While trying to find your way out you come across 4 others.

Each of them know 1 direction. There's two girls, they know east and west. There's also two boys, they know north and south. They are also from another country, and have different words for north, south, east, and west that you do not know.

You also know that one girl and one boy enjoys lying, and will lie to you at all times.

You may assume the following:

• You cannot ask a question that will silence them because they do not know. If they can't answer, assume they will kill you.

• They have no idea what they would say to a question, so "Are you about to say no?" or "What would you say if I asked you 'Is that way North?'".

• They have no knowledge of what the others in their group or gender would say to a question, or if they lie or not.

• They know enough English to understand your question, but not enough to respond in English.

• You cannot see the sun, moon, or stars. Pretend it's a cloudy night.

When you walk up to them they will hold their hands up in 2 directions, the correct way, and the opposite to the correct way. Can you ask them each a question, and be able to determine north, east, south, and west, if so, how?

You can ask them each a different question if you like, and label them BoyA, BoyB, GirlA, GirlB.

I in no way imply this is possible, or agree that it can be done. The point is to determine if you can, and if so how?

• So you are saying you dont know the answer, but want to see if someone in the community can find, am I right
– skv
Nov 15, 2014 at 13:41
• @skv Is that disallowed? Nov 15, 2014 at 13:42
• No... just asking if thats the case... and though they all have different words (From other languages) they can still understand english?
– skv
Nov 15, 2014 at 13:42
• @skv I edited that into the question. Nov 15, 2014 at 13:46
• If I ask a question with multiple answers, will the liars respond with a random false answer? (Just checking if probabilistic solutions might be an option.) Nov 15, 2014 at 15:31

The problem is insufficiently constrained to come up with a guaranteed solution.

We know that one boy and one girl are knaves, but we don't know whether the other boy and girl will tell the truth. They may or may not lie some or all of the time. The allowed assumptions (which do not include anything like "the other boy and girl always tell the truth") eliminate the usual ways of determining truth-telling value by disallowing asking questions about the other parties' answers. But those generally depend on knowing the distribution of orientations anyway (e.g. knowing there's a knight and a knave, but not who's who). Because of this, asking a single question to a single agent is insufficient to determine which of the pair is the knave. Both boys or both girls must be questioned.

If two same gendered agents give different answers to a question you know the answer to, that's all well and good. But if they don't, you don't know whether either of them will continue lying to you (considered as individuals; obviously at least one of them will). If they give different answers to a question you don't know the answer to, then you have no way of knowing who is the liar. This is the sticky wicket: when you ask a question, you know the answer or you don't. Either way, it's possible to fail to gain any information.

You can ask them each a different question if you like

If you're allowed to ask as many questions as you want, then you can build a probabilistic model. However, this isn't reliable either, because we can't assume the agents aren't malicious. For example, the non-guaranteed-knave might be a malevolent joker: lying until you're confident of knave status, then telling the truth when you ask about the directions. Probably because they got cranky about being asked so many questions.

If "a question" means only one question per agent, then you're basically down to a coin toss, since the boys show you north and south. This is because you need at least two different questions to determine a liar (granting for the sake of argument that doing so is possible): one with a known truth value and one to determine a direction.

You cannot see the sun, moon, or stars. Pretend it's a cloudy night.

I suppose you could just ask a question that would take sufficiently long enough to answer and wait for the sun to come up. But I assume the point of this puzzle is deducing the answer, not lateral thinking.

• I'll give others a chance to do better than this, if they can't I'll accept. Nov 15, 2014 at 17:09

You can say:

Can all of you point the cardinal point you know?
They won't answer cause they don't know English but they can move their arms to give the right answer. The 2 boys and the 2 girls will point only two direction, because two of them are lying. But after, if you ask Can all of you point the opposite cardinal point you know? you will easy find whom is lying and identify North, South, East and West.

• They know enough English to understand your question, but not enough to respond in English. They answer in THEIR language. When you ask them a question they out their 2 arms up and you have to ask which [direction] an arm is pointing in. Nov 15, 2014 at 14:22
• @warspyking Got it, they will talk to us anyway. I'll try to find a new solution editing this one. Nov 15, 2014 at 17:06

Ask the first girl whether 1+1=2. Now, the other girl is pointing both west and east, and you know whether she's a liar, so ask "Is [that direction] the same one that the sun rises in?"

• @Lopsy when you ask 1+1=2 how do you determine if she says yes/no when they respond in another language? Nov 15, 2014 at 14:23
• Oh, I thought they only had different words for directions. Anyway, I'll just ask "Define X to be the string 'yes' if 1+1=2, and 'no' otherwise; what is X?" Nov 15, 2014 at 14:27
• Edit that into the answer and I'll accept. Nov 15, 2014 at 14:49
• @warspyking This is not sufficient. See my answer for why. Nov 15, 2014 at 16:37

Well, I might be repeating after mr. Screen Name, but, analysing this logical problem, I've come to a conclusion on how bad your conditions are in such a situation. Let us imagine a much luckier situation than the one presented, killing some of the assumptions:

1. You can ask as many questions as you want to each of the kids.
2. They speak perfect English.
3. You can ask them whatever you want (and treat silence as a signal of ignorance), yet they still don't know each other's nature, knowledge or other things. Also, knaves will not react to a question they don't know answer to.

Now I believe that under these assumptions

This problem is still unsolvable.

And I'm going to prove it.

After you found a seemingly proper solution, you got out the jungle and into a village - there you saw an old chief. When you mentioned weird kids to him, he laughed and said:

- I hope you paid no attention to their answers. You wouldn't define any of the directions from them.
- But why so, - you say, - one of boys always tells lies! I could definitely exploit that. I used several neat strategies, like, asking them about their own answers, or whether 1+1=2, or, the sun-rise point, and, after some time I got proper evidence to follow the direction. Aren't I in the village right now?
- Yes you are, - said chief, - but it's a matter of luck. I think I have to tell you about siblings called BoyA and GirlA (it's their indian names).
They were raised pretty much in the same way as BoyB and GirlB, except for one strange thing in their perception - they suddenly remember wrong directions whenever it's dark outside. They also forget facts connected to it - for example, even if they saw sun rising from the sea during day, they will think that it rose from the forests as night comes - because...
- Because they think it's where the east supposed to be, - you interrupt, - I get the message. And why is it important to my situation?
- As I told you - BoyB and GirlB always lie to people, that's a fact. However, BoyA and GirlA always lie to people as well, the only problem being - they lie according to their perception. So, they would tell you 1+1 is not 2, yet they can accidentally point you the right south or east, thinking they are deceiving you. As I said - two always lie, and two, well... sometimes? I didn't guarantee anything on them.
- Let's make this straight - you say, - even after I found the way, I might suppose: "What if the way is different, and boys are swapped - thus, they would give me the same answer to every question, but I would be going in the wrong direction?"
- I'm glad we set this straight, - the chief answered. Also, I'm not the chief you talked to, I'm his twin from the cannibal village. Get in the cauldron.

The end.

You would approach the girls and ask them to point up if they know either directions the boys are pointing to, and point down if they don't. The girl pointing down is the truthteller. Then ask the truth teller to point in the direction she knows. Repeat with the boys. This way since they can understand what you say, you can reveal who is telling the truth without having to actually understand what they say. Also since there is the constraint that each person only knows one direction this works.

assume that only boys know north and south, and girls know east and west, and each gender of one of them lied, they will standing on the same place at the same time.

try to shake hands with them, determine if they are left-handed or right-handed.
ask boy which is north, if they are right-handed, they would use their right hand to point at the correct one.
while the wrong one always lie, they would try to point the right one, but then they might point to the wrong.
the one that didn't change the position will be right

I HAVE EDITED TO ANOTHER ANSWER

• This way you dont have to ask them at all wikihow.com/Find-True-North-Without-a-Compass
– skv
Nov 15, 2014 at 14:07
• Not what I had in mind... I'll have to edit the question slightly to fix more loopholes like this. Please delete. Nov 15, 2014 at 14:08
• sorry, thought that you need to find who lied Nov 15, 2014 at 14:18