Question from the Russian Olympiad. (Translated with Google Translate, with a translation fix.):
In a circle there are $181$ people, each of whom is either a knight or a liar (liars always lie, and knights always tell the truth). Each of those standing said: “Two places away from me there is at least one liar.”
Find the smallest possible number of liars among these $181$ people.
My solution:
I first estimated that there were $5$ people then the minimum number of liars is $2$. because $1$ said $3$ liars $2$ said $4$ liars $3$ said $5$ liars and $4$ said $1$ liars and $5$ said $2$ liars. then if $1$ liar then $4$ and $3$ knights. Whence it follows that $4$ is also a liar and $2$ knight. Well, then the minimum number of liars in our case is $90$.
Question:is it correct answer?
EDIT: I’ll try to explain it. Let's say if there are people in a circle with numbers $1,2,3,4,5,6$ …, $181$ then this means that #$1$ says that either #$3$ or #$180$ is a liar, #$2$ says that either #$4$ or #$181$ is a liar and so forth.
Best Answer
If I understand correctly, everyone in the circle says 2 persons away (in any direction) there is a liar.
The minimal pattern that matches is $KxLyK$ - the $L$ lies because there isn't one, and the $K$'s both tell the truth.
We can interleave two of those patterns and repeat pattern $KKLLKK$ 30 times.
Then you have to close the circle with a $L$. On both sides, the second item is a $K$, which repeats the pattern.
The solution therefor: $61$