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# I got Charlie's number

## Question

They wear numbers in the land of Truthtellers and Liars, and it was a great help. I had to find Bob, and I knew only that he was across the room talking with two of his friends. I decided on a direct approach. I asked the group, "Which of you is Bob?" The person wearing 576 replied, "I am." The person on his left, wearing 238, disagreed: "Not at all. It is I who am Bob." Since number 382 remained silent I prodded him: "You're not helping all that much by your silence. Might I trouble you for at least a clue?" He smiled slyly and ventured, "We're a strange group, we are, I would not expect to receive all that much help: only one of us will ever tell you the truth." "Thanks," I said. Next, I fixed my attention on Bob and told him: "I really need to find Charlie." I knew he was in the room elsewhere, and it held only two other small groups: 3117 was talking with 1137, and 4741 was standing next to 2305. Bob helpfully(?) motioned to one of the groups. Hmmm... should I believe him? Not only did I want to meet Charlie, I also wanted to know whether he would tell me the truth. I decided on the second pair, and blurted out my question "Does Charlie tell the truth?" Number 2305 replied (with a yes or no;) and with that I was able to deduce Charlie's number. But ... could I trust him?

What do we know about Bob's number and type, and about Charlie's number and type?

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I'm coming up with an answer that makes me think my analysis is just a little off.

Among the first group of three people, consider the case where 382 (the person who said there is only one truth-teller in the group) is a truth-teller and then consider the case where 382 is a liar.

If 382 is a truth-teller, then because his statement is truthful we know that the other two people in that group must be lying and therefore not Bob. So 382 is Bob and is truthful.

If 382 is lying, then the fact that his statement is false implies that either there are NO truth-tellers in the group or there are >1 truth-tellers in the group. If there are >1 truth tellers in the group and 382 is not a truth teller, then both of the other people in the group must be truth-tellers, but that's impossible since they both claimed to be Bob. So if 382 is lying, then everyone in the group must be lying. And that means that 382 must be Bob (because both other people who claimed to be Bob were lying) and Bob is a liar.

So we know that 382 is Bob, but I don't know whether Bob is a truth-teller or a liar.

If you somehow know that Charlie is among the second group of people that you talk to at the end of the puzzle (I'll talk about this assumption more later on), then the fact that you were able to deduce Charlie's number implies that 2305 answered "no" to whether Charlie tells the truth. If 2305 had answered that Charlie does tell the truth, then that doesn't tell you whether 2305 is Charlie or the other person with him is Charlie. If 2305 answers that Charlie does not tell the truth, then you know that 2305 cannot be Charlie because Charlie would not answer that way if he's a truth-teller and would not answer that way if he were a liar.

So you know that 2305 must have answered that Charlie is not a truth teller and that 2305 is not Charlie. But at this point, I don't know whether Charlie is a truth-teller or a liar, and I can only know Charlie's identity if there's some way of knowing that he's in the group of {4741, 2305}.

The fact that I can't solve the puzzle at this point actually tells me something. I initially thought that 382's statement that there is only one truth-teller among the "group" was referring to the "group" of three people that you're talking to at the beginning of the puzzle. That assumption leads to no solution (as far as I can tell). Alternatively, 382's "group" might have been referring to the entire room full of people. That ends up being a little different.

Suppose 382 is a truth-teller, and therefore the only truth-teller in the room. That would imply that 4741 and 2305 are both liars, and Charlie would have to be a liar because Bob is the only truth-teller. Remember that the fact that we know who Charlie is after hearing 2305's answer implies that 2305 answered that Charlie does not tell the truth. But in the scenario where 382 is the only truth teller (and is not Charlie), it would be impossible for 2305 to answer that Charlie is not a truth-teller because both 2305 and Charlie must be liars. So we can conclude that 382 cannot be a truth-teller.

So now we know that 382 is Bob and is a liar. If Bob said that Charlie was in the group of {4741, 2305} then we would know that Charlie is really in the group of {3117, 1137} but we wouldn't have any way of knowing which of those two he is. But if Bob said that Charlie was in the group of {3117, 1137} then we would know that Charlie is really in the group of {4741, 2305}, and based on the fact that 2305 must have said that Charlie is not a truth teller we would know that Charlies must be 4741. Since the puzzle says that you now know who Charlie is, that implies that the second scenario in this paragraph must be the one that happened. Charlie is 4741. And I don't know whether Charlie is a truth-teller or a liar, I only know that he's the opposite of whatever 2305 is.

When I did my first analysis, I was only able to tell that 382 must be Bob because I was working on the assumption that the "group" he was referring to was the group of three people you were talking to at first. If the "group" was actually everyone in the room, then it's possible that 382 is a liar, and that one of the other people in the group of three was actually telling the truth and is Bob because there could be another truth-teller somewhere in the room to make 382's statement a legitimate lie. The puzzle says that you know who Bob is after hearing 382's answer, so that seems to make it impossible to work under the assumption that the "group" refers to everyone in the room.

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The group 382 mentioned was his group of three friends.
I meant that to be a given, but you also correctly deduced it.

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This is partially solved, but there are four questions to answer.

What can be known about Bob's and Charlie's number and type?

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I don't know about everyone else, but I'm stuck.

If the "I" in this problem has information that we (the problem solvers) are not privy to by virtue of the OP, then I could come up with a solution.

If the "I" in this puzzle definitely does NOT have any more information than what's presented, then I am Stuck with a capital S.

Because otherwise I can't rule out the possibility that Charlie is anyone (except Bob and 2305) and I can't make sense of the statement that you've figured out which number Charlie has. As far as I can tell, if 2305 is telling the truth then Charlie could be in either of the two remaining groups as long as he's a liar, and if 2305 is lying then he could still be in either of the two remaining groups if Charlie is a truth-teller. Telling for sure which of the two remaining groups of two Charlie is in, and thereby being able to draw a firm conclusion about 2305's statement that would let you deduce who Charlie is, seems to require more information than what was presented to us.

Unless that's the whole point of the problem. If somehow the "I" in this puzzle had info that let him know that Charlie was in the group of 2305 and 4741, then he could deduce that Charlie is 4741 if 2305 answered "no" (which is the only answer that would produce a solution), but would not know if Charlie is a truth-teller or a liar.

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You are on the right track. The logic can be tightened up,

but you basically have the correct information.

382 says that at most one of the three tells the truth.

Suppose 382 lied when he said that.
Then the truth of the matter is that at least two tell the truth.
By supposition, 382 is not one of them. So both 576 and 238 tell the truth.
Since they contradict each other, this cannot be the case.

Therefore 382 told the truth.

Person 382 must then be the one-and-only truth-teller.
Since 576 and 238 both lied, neither of them is Bob.

Bob is 382, and Bob is a truth-teller.

I now tell Bob I need to find Charlie, and Bob nods toward one of the other groups.
Knowing Bob is a trusted informant, I logically go to the group he indicates.
The OP indicates that group comprises 2305 and 4741: according to Bob, one of them is Charlie.

I ask them: Does Charlie tell the truth?
Number 2305 replies with a Yes or a No.

The OP now gives us this piece of information:
From his answer I am able to deduce Charlie's number.

Suppose 2305 replies Yes. Meaning: 2305 claims that Charlie tells the truth.
It is possible that 2305 is a truth-teller. And if so, Charlie does in fact tell the truth.

Then, since by supposition 2305 can be a truth-teller, 2305 could be Charlie.

But it's also possible that 4741 is a truth-teller. So it's also possible that 4741 is Charlie.
So if 2305 replies Yes, I can't deduce Charlie's number.

But OP tells us that 2305's answer allows me to do that.
Therefore 2305 does not reply Yes.

Therefore 2305 replies No.

Suppose 2305 is a truth-teller. Then Charlie does not tell the truth.
Charlie then cannot be 2305, and must be 4741.

Suppose 2305 is a liar. Then Charlie does in fact tell the truth.
Charlie then cannot be 2305, and must be 4741.

Either way, Charlie is 4741. And that is all we know.

So, regarding Bob's number and type and Charlie's number and type we can say:

1. Bob is 382
2. Bob tells the truth.
3. Charlie is 4741
4. Charlie's truth type is unknown

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Giving the solution to plasmid.

See previous post for the reasoning.

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I see what tripped me up. I interpreted 382's statement as "exactly one of us tells the truth", and not "no more than one of us tells the truth". That would allow 382 to be a liar if all three of the initial group were liars.

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It's not as precisely written as it might be.

I intended his statement to be "... (one and) only one of us will ever tell you the truth."

I think the words in red are normally taken as implied; but making things explicit is always a good thing.

The intended implication is that exactly one of the three tells the truth.

Which is needed to identify him as a truth-teller -- which is needed to proceed with the second part.

That is, you need to know that they can't all three be liars.

The first statement in the Solution leaves the one and only one condition unstated, as well.

But the solution needs the one and only one condition.

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Doesn't the "one and only one of us will ever tell you the truth" interpretation mean that it's possible for all three of the people in the first group to be liars?

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Doesn't the "one and only one of us will ever tell you the truth" interpretation mean that it's possible for all three of the people in the first group to be liars?

Yes it does. Bob is a truth-teller.

We use Bob's truthfulness to determine which group Charlie is in.

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