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At a basketball fair, everyone is a fan of just one team (Celtics, Lakers, or 76ers). Celtics fans always lie to Lakers fans, Lakers fans always lie to 76ers Fans, and 76ers Fans always lie to Celtics fan. Other than this, everyone always tells the truth. Everyone else has been going to this fair for 20 years now so they all know which team others support. This is my first time at the fair so I don't know anybody's affiliations, however, because I am wearing a Celtics shirt, everybody else knows that I am a Celtics fan.

What question should I ask someone to determine if they are a Lakers fan?

Let's say you overhear a conversation between two people. Does a single statement exist such that upon overhearing it, I can determine whether the two people support the same basketball team?

  • This is not clear. First of all, are you assuming that everyone is a fan of exactly one of these three teams? Secondly, since you say "Everyone...always knows what type of fan someone is" why do they need to ask any questions at all? What does that assumption mean if not "everyone knows who is a C-fan, L-fan, Sixers-fan"? – lulu Feb 01 '21 at 14:34
  • For your first question, yes everyone is a fan of just one team. And for the second part, it means that if I ask someone a question, they will know which fan I am and say the truth/lie accordingly. I guess in this case, I myself do not know what type of fan anybody is but everyone else does (maybe because everyone has been going to this convention for years and they all know each other but this is my first time going so I have no clue) . –  Feb 01 '21 at 14:37
  • This needs editing, the assumptions are not at all clear. If you want to say that everyone is a fan of exactly one of these three specific teams, that needs to be made explicit. But, more important, you need to say exactly who knows what in each conversation and to whom any given statement is addressed. That is, if I am a Celtics fan but I don't know which team you favor, then I have no idea whether I should lie to you or not. – lulu Feb 01 '21 at 14:40
  • I just edited it for clarity. –  Feb 01 '21 at 14:45
  • Just to say: if you are restricted to binary (T/F questions) then, whatever you ask, you can only get one of two bits of information so you can only distinguish between two states, thus it is impossible. If, on the other hand, the answers can be more complicated then I think you need some rule to clarify how the liars choose between multiple possible lies. – lulu Feb 01 '21 at 14:46
  • The questions must all be yes/no. In your above statement, what would be impossible? –  Feb 01 '21 at 14:47
  • To distinguish between more than $2$ states. Whatever you ask, the person can only respond Y or N. How could you hope to identify three possibilities with only those two responses? – lulu Feb 01 '21 at 14:49
  • I think it'd depend on the logic between the true and false. As in part 1, in the below answer, with just 1 question, we were able to distinguish between 3 possibilities. –  Feb 01 '21 at 14:50
  • That doesn't make sense. A Yes tells you which team? A No tells you which team? That's at most two teams you have identified. – lulu Feb 01 '21 at 14:51
  • We have to distinguish between only 2 states; whether the person is a Lakers fan or not. – EBP Feb 01 '21 at 14:53
  • So in the below answer, the question being asked is "Are you not a Celtics fan." If they are a Celtics fan, they will say no. If they are a Lakers fan, they will say yes. If they are a 76ers fan, they will lie and thus say no. So, if the response that I get back is "no," then they are not a Lakers fan, but if the response I get back is "yes," then they are. –  Feb 01 '21 at 14:53
  • Yeah and for part b, the two states would be: Do they support the same team or no. –  Feb 01 '21 at 14:55
  • do you hear statement and response for the second question or just the statement? – lulu Feb 01 '21 at 14:55
  • just a statement. For example, if you overheard "I am a Lakers fan," then either the person saying it was a Lakers fan and the other person is either a Celtics or Lakers fan or the person saying it is NOT a Lakers fan. In that case, they can either be a Celtics fan lying to a Lakers fan or a 76ers fan lying to a Celtics fan. –  Feb 01 '21 at 14:57
  • The simple statement "You just lied to me!" does the job. – lulu Feb 01 '21 at 14:58
  • why? If I'm talking to someone of the same fanbase as me, they will not lie to me. I cannot say "you just lied to me" because they didn't lie and I cannot lie. –  Feb 01 '21 at 15:00
  • I assumed you heard part of an ongoing conversation. Easily modified, as in "you are going to lie to me." – lulu Feb 01 '21 at 15:01
  • Again, that statement will not work. If I hear that, it just tells me that the two people are fans of different teams. It will not tell me that they are fans of the same team. Because if they were, such a statement could never arise as both people are bound to truthfulness. –  Feb 01 '21 at 15:03
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    But any single statement has only one interpretation. That's why I asked if you also heard the response. – lulu Feb 01 '21 at 15:09

1 Answers1

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For the first question, you can ask whether the person in question is a Laker-or-76er fan. If it is a Celtic fan, he answers no and you know that he isn't a Laker-or-76er fan. If it is a 76er fan, he will lie and tell that he isn't a Laker-or-76er fan, so he tells you he's a Celtic fan essentially. If you ask the question to a Laker, he'll answer positive, and hence you know for sure that he is a Laker.

As for the second question, there does not exist a single statement such that we always know whether the two are fans of the same club or not. If two lakers fans are chatting and we overhear some statement $\varphi$ such that we immediately know they are fans of the same club, then this does have nothing to do with the two actually being fans of the same club, but only with our ability to do some logical deduction, hence we will always deduce the same conclusion. Or, as summarized in Lulu's comment; every sentence has only one interpretation.

EBP
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  • for the second question, I don't think it fully answers the question, though. That statement only tells me that they are not fans of the same team. We need a question that will confirm that they are part of the same team. –  Feb 01 '21 at 14:58
  • I didn't quite get your reasoning for #2. If two Lakers fans say something and I can deduce from it that they are fans of the same team, why does it mean no such statement exists? –  Feb 01 '21 at 15:12
  • Because the deduction itself is independent of the clubs they are fan of. – EBP Feb 01 '21 at 15:15
  • Is my understanding correct: WLOG, let the person whose statement you overhear be a Celtics fan. If they said a true statement, then the other person is etiher a 76ers fan or a Celtics fan. If they said a false statement, then the other person must be a Lakers fan. There is no way to actually see whether they are the same type of fan. –  Feb 01 '21 at 15:37