How many liars are in the room?

Question

There are five people in a room. Each person is either a knight, who always tells the truth, or a liar, who always lies.

Each person is asked the following question:

How many liars are among you?

The answers are: "one", "two", "three", "four", "five".

Missing piece of riddle: each of them knows what kind the others are. I failed to solve due to not assuming that. — Joshua, Sep 25 '17 at 16:10
@Joshua I think it's implied because the knights always tell the truth, it's not just that they think they are telling the truth. — RothX, Sep 25 '17 at 16:29
What kind of entity is asking the question, and would all five of the "people" in the room assume "you" refers to all five collectively? If there were a group of three people in one corner who answered "one", "two", and "four", and a group of two in the other corner who answered "three" and "five", there might be three liars in the first group and two in the second (five total). — supercat, Sep 25 '17 at 20:22
@Joshua The fact that you got five answers demonstrates that. — David Schwartz, Sep 25 '17 at 22:16
@DavidSchwartz: Alternate solution. They're all liars and nobody knows any of the others are liars. — Joshua, Sep 25 '17 at 22:35
@Joshua Then the one who said there are five liars was not lying. A lie is, by definition, a false statement made with knowledge of its falsehood. We are given that each person in the room either always lies or always tells the truth. — David Schwartz, Sep 25 '17 at 22:49
... but what if the liars tell the truth to lie about being liars? That'd mess up the whole thing! — Dirk v B, Sep 25 '17 at 22:50
@DavidSchwartz: But he knows he does not know and yet claims it. That is yet a lie. — Joshua, Sep 26 '17 at 02:17
How would any of them know how many liars are in the room? A good liar has a good enough lie not to be detected. Normally you can't enter a room, and know who is lying, unless the statement is completely absurd. The premise assumes they know for certain how many liars are in the room. — cybernard, Sep 26 '17 at 03:17
@RothX Is it common among logic puzzles that "always tells the truth" implies omniscience? I'm not an expert puzzler, but I would assume "always tells the truth" would by default mean "to the best of their knowledge," unless additional clarifications are given. — GrandOpener, Sep 26 '17 at 18:23
@GrandOpener I don't know if I'd say omniscience is assumed necessarily, but it's often assumed that everyone involved is a "perfect logician," meaning they can solve any logical problem correctly. Which sort of implies they will never say anything is true if it is actually false. Many problems of this type give people the option to "pass" or say "I don't know." Since that isn't the case here, it's reasonable to assume omniscience, or at the very least omniscience about everything relevant to the situation. — RothX, Sep 26 '17 at 22:27
@GrandOpener It only implies omniscience of the person composing the puzzle of the truth or falsity of the claims made within the puzzle. Of course that's the case, it's their puzzle. — David Schwartz, Sep 27 '17 at 04:29
@DavidSchwartz But it's not only that. In line with what RothX explained to me, "Knights always tell the truth" taken at face value implies not only that the creator of the puzzle knows what is truth--it must also imply that the knights always correctly know what is truth, else they couldn't always tell it. — GrandOpener, Sep 27 '17 at 17:54
@GrandOpener That's nonsense. The person who makes the puzzle knows what the various people in it say. And it is the person who makes the puzzle who is telling you that every character in it either always tells the truth or always lies. The "knight" might have no idea what is true or might even think he's lying but, as it happens, he winds up telling the truth, as determined by the person making the puzzle. That's what makes him a "knight" -- within the context of the puzzle, everything he says happens to be true. There are no actual knights who know things. — David Schwartz, Sep 27 '17 at 19:00
@DavidSchwartz Okay, I think I understand the semantics of your viewpoint now; it was unfortunate that I focused on the knights. About the liars: the liars always lie. Since we established that a lie must be intentional, that necessitates that any liars who exist must all individually know the total number of liars, else they would not be able to lie about how many liars there are, correct? And the semantic difference you present is that "tell the truth" need not be intentional, so the knights could simply be good guessers (etc.), and we cannot conclusively say what they "know"? — GrandOpener, Sep 28 '17 at 19:04
@GrandOpener Yes, that's what I'm saying. And the puzzle maker assures us that all this is true. — David Schwartz, Sep 29 '17 at 20:04

score 43 · Accepted Answer · edited Sep 25 '17 at 18:19

43

I am sure there are :

Four liars.
Only the fourth person is telling the truth.

if there is 1 liar, there must be 4 answers saying there is 1 liar
if there are 2 liars, there must be 3 answers saying there are 2 liars
if there are 3 liars, there must be 2 answers saying there are 3 liars
if there are 4 liars, there must be 1 answer saying there are 4 liars (this is the actual case)
if there are 5 liars, no one can say so, or they would be telling the truth.

edited Sep 25 '17 at 18:19

Daniel

103
3

answered Sep 25 '17 at 07:27

Jamal Senjaya

17,864
2
41
147

3

There could be 5 liars if the truth is "I don't know", but such possibilities are usually ignored by this type of question. – Brilliand Sep 25 '17 at 18:09
1

Hi @Jamal Senjaya, I agree with David K. Is it not more correct to say that we can definitively rule out five liars? This is because if there were five liars, as none of then can tell the truth, nobody could have given the answer "five" above. – MikeRoger Sep 26 '17 at 08:35
@brilliand you could also read the question to mean "How many liars are there, at least, among you?". – Clearer Dec 14 '17 at 11:43

score 26 · Answer 2 · answered Sep 25 '17 at 15:49

There are four liars

This is the same answer as others have already found, but I would like to provide what I see as a better-worded explanation:

Since each person claims there are a different number of liars, there can be at most one person telling the truth (any two people telling the truth would be contradicting each other). Further, at least one person must be telling the truth, since if no-one were telling the truth, then the person who claimed there were five liars would be telling the truth, resulting in a contradiction. Thus there is exactly one person telling the truth, and hence four liars.

(Note that the puzzle doesn't actually require knights who always tell the truth and liars who always lie: it would work equally well if "liar" was understood to mean "person who lies in their claim about how many liars there are in the room".)

This is the path I took as well, it seems a simpler way to get there. — Dan Henderson, Sep 25 '17 at 18:59

score 9 · Answer 3 · edited Sep 25 '17 at 09:19

Here is my logic:

If there are n liars then there must be 5-n people who must speak truth and since the number of liar in the room is a unique number it cannot have different values there can only be one person who speaks the truth because if we consider any two of them are speaking truth then we get two different no of liars in the room which is not possible rather its unique. So only one person can be speaking truth and all other must be liar So,there are 4 liars in the room.

How many liars are in the room?

3 Answers3