# How many liars are in the room?

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".

How many liars are in the room?

• 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
• Depends. Am I in the room? – MikeTheLiar Sep 25 '17 at 20:34
• @Joshua The fact that you got five answers demonstrates that. – David Schwartz Sep 25 '17 at 22:16

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.

• 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
• 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 8:35
• @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

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

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.