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There are two types of people on an island: truth tellers and liars. Truth tellers always tell the truth, while liars always lie. A foreigner meets six of the inhabitants, and asks each of them the following question: "How many of you are truth tellers?"

The first five responses were the following:

"Two of us are truth tellers",
"None of us are truth tellers",
"Three of us are truth tellers",
"Only one of us is a truth teller",
"Three of us are truth tellers".

This was of course not sufficient for the foreigner to know who is a truth teller and who is a liar. However, when he heard the final answer, he knew right away how many truth tellers there are.

What was the final response? How many truth tellers were there?

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  • $\begingroup$ Should we be interpreting "N of us are..." as "There are at least N ... among us", or as "Exactly N of us are ..."? $\endgroup$ – Jonathan Allan Sep 8 '16 at 20:02
  • $\begingroup$ @JonathanAllan I'm pretty sure the problem wouldn't be solvable with the former interpretation, but it is with the latter :-) $\endgroup$ – Rand al'Thor Sep 8 '16 at 20:03
  • $\begingroup$ @randal'thor Agreed it makes most sense with the question posed too $\endgroup$ – Jonathan Allan Sep 8 '16 at 20:15
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    $\begingroup$ It should be interpreted as Exactly N of us are :) $\endgroup$ – Gat Sretcarahc Sep 8 '16 at 20:28
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    $\begingroup$ The final answer must also be in a form of "exactly N of us are truth-tellers". $\endgroup$ – Gat Sretcarahc Sep 8 '16 at 20:38

10 Answers 10

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For N truth tellers exactly N people will say "there are N truth tellers" and for the problem to be solvable there must be exactly one N. So let's consider each possible N:

0: 1 person says it -> false
1: 1 person says it -> true if the last person says something else
2: 1 person says it -> true if the last person says it
3: 2 people say it -> true if the last person says it
4: 0 people say it -> false
5: 0 people say it -> false
6: 0 people say it -> false

Now we have 3 possible answers

But if the last person says 2 or 3 there are two valid N's (1 and whatever he says) making the problem is unsolvable.

So

There is one truth teller
and the last person says some number other than 1, 2 or 3.

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  • $\begingroup$ So clear and concise - no rambling just the necessary details, doesn't monkey about with swathes of text to describe what can be summed in a few simple, clear, to the point, redundancy skipping, no frills and poignant phrases that do all the work of a gallery of pictures in a city of pottery museums. Does not leave us wondering what else or what if, it satisfies us enough to not be left wanting or begging... +1 $\endgroup$ – Jonathan Allan Sep 9 '16 at 14:34
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We have five statements to process:

  1. "Two of us are truth tellers".
  2. "None of us are truth tellers".
  3. "Three of us are truth tellers".
  4. "Only one of us is a truth teller".
  5. "Three of us are truth tellers".

These five statements are all mutually contradictory except 3) and 5). So out of the first five speakers, either none, one, or two are telling the truth.

  • If none of the first five speakers are telling the truth, then in particular both speaker 2) and speaker 4) are lying, so the number of truth-tellers among all six cannot be zero or one. Contradiction.

  • If two of the first five speakers are telling the truth, then it must be speaker 3) and speaker 5), so speaker 6) must be the third truth-teller, so speaker 6) must also say "Three of us are truth tellers". However, if speaker 6) does say this, then as far as the foreigner knows, it might be speaker 4) and nobody else telling the truth. This contradicts the assumption that speaker 6)'s statement is enough for the foreigner to know how many truth-tellers there are in total.

So exactly one of the first five speakers is telling the truth. That means speakers 2), 3), and 5) are lying, so the truth-teller among them must be either 1) or 4).

  • If it's 1), then speaker 6) must be the second truth-teller, so speaker 6) must also say "Two of us are truth tellers". However, as before, if speaker 6) does say this, then as far as the foreigner knows, it might be speaker 4) and nobody else telling the truth. Contradiction.

So speaker 4) is telling the truth, which means

there is exactly one truth-teller in total.

Now what must speaker 6)'s response have been? It must be a lie, so it can't be "Only one of us is a truth-teller". It also can't be "Two of us are truth-tellers" (since then as far as the foreigner knows, it could have been speakers 1) and 6) telling the truth) or "Three of us are truth-tellers" (since then as far as the foreigner knows, it could have been speakers 3), 5) and 6) telling the truth). Any of the other options seems to be possible. So speaker 6) could have said

"None of us are truth-tellers" or "Four of us are truth-tellers" or "Five of us are truth-tellers" or "All of us are truth-tellers".

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  • $\begingroup$ I wonder if it could be written differently so that we really do know that we don't know after the 5th statement. $\endgroup$ – John Sep 8 '16 at 20:08
  • $\begingroup$ So, you're most certainly correct. My problem is that if we operate with the understanding that the 6th answer will remove all doubt, we could've skipped all this and said, he cannot answer 3 because it adds ambiguity, he cannot answer 2 because it adds ambiguity. In other words, "they must give us an answer that does not add confusion". If they do, we know they are the liar. Person 6 was in the unenviable position of being deemed a liar before they could answer. $\endgroup$ – John Sep 8 '16 at 21:08
  • $\begingroup$ We know the sixth person will be a liar, but only after being told hearing his answer will lead to knowledge also number six on the island says "I am not a number I'm a free man" $\endgroup$ – user19641 Sep 8 '16 at 22:52
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    $\begingroup$ Your answer is, of course, correct (although I approached it a different way). But I don't understand your first bullet, which I read as, "If all of the first five people are lying, then two of them in particular are lying, so the number of truth-tellers among all six must be at least two." Huh? $\endgroup$ – Peregrine Rook Sep 8 '16 at 23:03
  • $\begingroup$ @PeregrineRook No: no two of the first five statements can be true, except possibly 3) and 5), so at most two of the first five speakers are telling the truth. $\endgroup$ – Rand al'Thor Sep 8 '16 at 23:06
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There is one truth teller.
The final response was "Five of us are truth tellers."

This is because if the final answer were one of the already chosen numbers, the foreigner would not be able to distinguish the truth from a lie.

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    $\begingroup$ Can you expand on this answer a bit? I don't understand your non-spoilertagged sentence. Also, why can the final response not be e.g. "Four of us are truth-tellers" or "None of us are truth-tellers"? $\endgroup$ – Rand al'Thor Sep 8 '16 at 19:57
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    $\begingroup$ It could have also been "Six of us are truth tellers." I think the point is that it was not one, two, or three. $\endgroup$ – Ian MacDonald Sep 8 '16 at 20:07
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Person Two Must be lying, his statement contradicts itself.

Person Three and Person Five are either both lying, or both telling the truth, along with Person 6.

Person One is either lying, or telling the truth along with Person 6.

Person Four is either telling the truth alone, or lying.

If Person 6 is lying, then Person 4 tells the truth, however, any lie makes this the case, and the Question includes guessing Person 6's statement.
Person 6 must say something that is true when Person 4's statement is true, otherwise the outcome remains ambiguous.
The realm of things Person 6 can say that are unambiguous and possible are those that match 1 AND 2 AND !3 or 1 AND 3 AND !2.

My guess Person 6 said

An odd number of us are Truth Tellers

Because it fits the statement pattern best. This leads us to exactly

Three Truth Tellers, persons 3, 5 and 6.

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  • $\begingroup$ It took me a while to come around to this, but I believe that this answer works. But (1) I don't see how it fits the statement pattern best, and (2) the OP said, 'The final answer must also be in a form of "exactly N of us are truth-tellers".' But since he said that in a comment, and not in the question itself, it doesn't count. You're in the clear until the question is edited. $\endgroup$ – Peregrine Rook Sep 8 '16 at 22:43
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Rand al'Thor has the only real solution if you're looking for a single answer. Depending on interpretation though, the answer might be:

Multiple answers, or not enough information. It depends on how you want to interpret this sentence: "However, when he heard the final answer, he knew right away how many truth tellers there are." Normally this (kind of) sentence would indicate that the 'final answer' gave some all the information one needed for a solution.

...

But, if this puzzle has only one answer, then we have to make what's suggested in the sentence itself an additional premise. We'd need to say, "Oh, because he knew right away, then the answer cannot be any answer that would lead us astray of an answer that is already before us". And the only answer then, is one.

...

If the 'final answer' itself was used, alone, in determining the truth-teller count, it wouldn't be enough information to come to a single number of truth-tellers. There could be multiple answers in that situation. So, if we were the person on that island, we would not know the answer based solely on the responses.

That said, good first puzzle. Got me thinking.

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  • $\begingroup$ No, because if that's the last answer, then it would also be logically consistent for the 4th speaker and nobody else to be a truth-teller, so the foreigner wouldn't have been able to tell immediately how many truth-tellers there were. $\endgroup$ – Rand al'Thor Sep 8 '16 at 19:54
  • $\begingroup$ @randal'thor, Thinking, but, I'm kind of just going on an assumption that the PO is stating part of the premise when they say, we don't know until the final person has answered. $\endgroup$ – John Sep 8 '16 at 19:58
  • $\begingroup$ The foreigner doesn't know the answer until the final person has spoken, because the first five statements aren't enough on their own. (We could have worked this part out logically without having to be told it.) The really important piece of information is that he does know the answer as soon as the final person has spoken. And that's what invalidates the possibilities in the first spoilertag in the current revision of this answer :-) $\endgroup$ – Rand al'Thor Sep 8 '16 at 20:23
  • $\begingroup$ @randal'thor, yeah. I gotcha. I'm still thinking. $\endgroup$ – John Sep 8 '16 at 20:25
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So, if I understand this correctly:

(I am assuming that each answer means "exactly N of us are truth-tellers")

Because of answer 2, which is contradicting when true, we can conclude there is at least 1 truth-teller. When considering the answers, there are some different possible last answers with non-contradicting solutions

First:

1. "Two of us are truth-tellers" -> either one or two could be telling the truth

Then:

2. "Three of us are truth-tellers"-> either one or three could be telling the truth

Finally:

3. All other answers that are not "One of us is a truth-teller" -> exactly one is telling the truth.

Therefore:

Since the question specified that he knew for certain after the sixth answer, it has to be "None", "Four", "Five" or "Six of us are truth-tellers" and it evaluates to one truth-teller.

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A number of people have reasoned that the answer must be

one.

However, as long as it isn't specified how exactly the sixth person answered the question, there's some room for alternative solutions. If we allow the sixth person's response to be a statement that doesn't directly give a definitive answer to the question, different solutions can be found. For example, if the sixth person's response was

"There aren't three truth tellers."

then the answer would be

two truth tellers. If there were no truth tellers or a single truth teller, then the sixth statement would have to be a lie, resulting in a contradiction. If there are three or more truth tellers, a contradictory statement would have to be true. Thus, the only possibility is two truth tellers, those being the person who claimed there are two truth tellers and the sixth person who said there aren't three.

If the sixth person's response was

"There's either one or three truth tellers."

then there has to be

three truth tellers. Zero truth tellers contradicts the person who said everyone is a liar. One truth teller requires the sixth statement to be false, which is a contradiction. Two truth tellers and four or more truth tellers require contradictory statements to be true.

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  • $\begingroup$ You are correct (and other answers here have also taken advantage of this loophole/ambiguity), but the OP has specified that the sixth person's answer must be in the form "Exactly N of us are truth-tellers" for some N. $\endgroup$ – Rand al'Thor Sep 9 '16 at 10:21
  • $\begingroup$ But, in spite of my comment on Sconibulus's answer (with which this answer overlaps), the OP still hasn't edited that constraint into the question, and so I stand by my statement that it doesn't count. $\endgroup$ – Peregrine Rook Sep 9 '16 at 15:25
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If the responses are to be taken at face value

The final response is

"Two of us speak the truth"

making the answer

speaker 1 and 6 speak the truth

because

the final speaker says "Two of us speak the truth" immediately making only those two statements true. Had the final speaker said "Three of us", the first statement would become true as it did not specify "only two" as the fourth speaker does. The title also mentions the island only has 6 inhabitants, thus eliminating the possibility of any non-respondent truth-speakers.

However, if each speaker is defining exactly how many speakers are truth tellers, then

the final response is

"Three of us speak the truth"

making the answer

3, 5, and 6 speak the truth

because

these are the only three that responded with "exactly 3 truth tellers"

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  • $\begingroup$ Sorry, this answer is incorrect. In both cases, how would the foreigner know that it wasn't the 4th speaker telling the truth and there was only one truth-teller among the six? $\endgroup$ – Rand al'Thor Sep 8 '16 at 22:53
  • $\begingroup$ you're right. apologies :/ $\endgroup$ – Aaron P Sep 8 '16 at 23:00
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Let's rewrite the question as follows (I think best in positives...):

! A: "4 of us are liars"
B: "6 of us are liars"
C: "3 of us are liars"
D: "5 of us are liars"
E: "3 of us are liars"

Now let's figure what we can initially:

! B can't be truthful; being so would mean he's a liar, which would be contradictory
If D were truthful, it would mean D is the only truth-teller (even F would be a liar)
If A were truthful, it would mean A is one of two truth-tellers. B, D, C and E contradict A, so if A were truthful, the mysterious F would have to agree with A
C and E are either both truthful or both liars. If they are truthful, it would mean B, D and A are liars so F must be truthful (and agree with C and E)

From that I conclude multiple possibilities:

! If F says "3 of us are liars" then C, E and F are truthful only.
If F says "4 of us are liars" then A and F are truthful only.
If F says anything else, then D is truthful only.

So...:

! F can say "3 of us are truth-tellers" (Truthies: C, E, F; Liars: B, D, A)
F can say "2 of us are truth-tellers" (Truthies: A, F; Liars: B, D, C, E)
F can say anything else (Truthies: D; Liars: B, A, C, E, F)

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  • $\begingroup$ You rephrased the statements for no obvious reason (it's quite straightforward to solve the problem using the original wording) and then you labeled them A, B, C, D, and E in what appears to be a random order (I eventually figured it out). It hurts my brain to read your answer beyond the first paragraph, but I skipped to the end, and I see that it's wrong. $\endgroup$ – Peregrine Rook Sep 8 '16 at 22:54
  • $\begingroup$ @PeregrineRook I reworded it because it's easier for me to think in truths than in lies $\endgroup$ – Altainia Sep 8 '16 at 23:20
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If 0 of them are truth-tellers, then all of them must tell lies, so all of them must say something different than "0 of us are truth-tellers." Since 1 of them says this, this cannot be the truth, because then we would have a liar telling the truth. If 6 of them are truth-tellers, then all 5 statements that we have would need to agree. They do not all agree, so it cannot be that 6 of them are truth-tellers. If 5 of them are truth-tellers, then at least 4 of the 5 statements that we have would have to agree, but at maximum 2 of our 5 statements agree. So it cannot be that 5 of them are truth-tellers. If 4 of them are truth-tellers, then at least 3 of our 5 statements would have to agree, but at maximum 2 of our 5 statements agree. So it cannot be that 4 of them are truth-tellers. So this means the number of truth-tellers must be 1, 2 or 3.

We know that the nature of the 6th person's response is such that it allows the foreigner to deduce the number of persons. If there were 2 or more responses that the 6th person could give that would each allow the foreigner to deduce a unique answer, then we could not answer the question ourselves. So if there are 2 or more responses that the 6th person could give, then they all must lead to the same answer. Let us find out what can be deduced from each response that the 6th person could give. 1. Let's say the 6th person answers "0 of us are truth-tellers." We already know 0 can't be the answer. The only other options are 1, 2 or 3. In this scenario, the number of truth-tellers cannot be 2, because there are not exactly 2 people saying "2 of us are truth-tellers." The number of truth-tellers cannot be 3 because there are not exactly 3 people saying "3 of us are truth-tellers." So the only option left is 1, and we find no contradiction; there is indeed exactly 1 person saying "1 of us is a truth-teller." So if the 6th person's response is "0 of us are truth-tellers," then we can deduce that there is exactly 1 truth-teller. 2. Let's say the 6th person answers "1 of us are truth-tellers." Then there would be 2 people saying "1 of us are truth-tellers." So there cannot be exactly 1 truth-teller, because then one of these 2 people would be a liar and he would be telling the truth, and liars cannot tell the truth. But there cannot be 2 truth-tellers in this scenario either, because there would not be 2 people saying "2 of us are truth-tellers," and there cannot be 3 truth-tellers, because there would not be 3 people saying "3 of us are truth-tellers." But this means that if the 6th person answers "1 of us is a truth-teller" then we have a paradox where no answer is correct. So the 6th person simply cannot respond "1 of us is a truth-teller." 3. Let's say the 6th person answers "2 of us are truth-tellers." Then there would be only 2 possibilities: 1 or 2, because there are not 3 people saying "3 of us are truth-tellers" so 3 is impossible. But there are exactly 2 people saying "2 of us are truth-tellers" and there is exactly 1 person saying "1 of us is a truth-teller". So the foreigner cannot decide. Since we are told that the foreigner CAN decide, the 6th person could not have responded "2 of us are truth-tellers." 4. Let's say the 6th person answers "3 of us are truth-tellers." Then the number of truth-tellers could not be 2, because there would not be exactly 2 people saying "2 of us are truth-tellers." But again, there would be exactly 1 person saying "1 of us is a truth-teller" and exactly 3 people saying "3 of us are truth-tellers." So if the 6th person responds "3 of us are truth-tellers," then the foreigner cannot decide. But we are told that the foreigner can decide. So this is not a possible response. 5. Let's say the 6th person answers "4 of us are truth-tellers." Then the number of truth-tellers cannot be 2 or 3 because there are not, respectively, exactly 2 or 3 people saying "2 of us are truth-tellers" or "3 of us are truth-tellers." So there must be exactly 1 truth-teller, because that is the only option that cannot be disproven. 6. If the 6th person answers "5 of us are truth-tellers" or "6 of us are truth-tellers" then we can use the same reasoning as in case 5 to prove that the only option we cannot disprove is there being exactly 1 truth-teller. So in conclusion, the 6th person cannot respond "1 of us is a truth-teller" because that leads to no answer being correct. It cannot respond "(2 or 3) of us are truth-tellers" because that would lead to the foreigner being unable to decide the correct answer. It can respond "(0, 4, 5, or 6) of us are truth-tellers" and each of these answers allows the foreigner to conclude that there is exactly 1 truth-teller.

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