We have $n$ numbers $a_{1,\ldots,n}$ on a circle, each number is either $1$ or $-1$. $n> 6$
We define product $P_i$ = $a_i a_{i+1}\cdots a_{i+5}$. Here the subscripts are cyclic, so $a_{n+1} = a_1$
What's the minimal value of $S = P_1 + \cdots + P_n$. Express the answer in $n$
I tried to think of some special cases like $n=3$, finding minimal value of $a_1 a_2 + a_2 a_3 + a_3 a_1$. Obviously then there are at least one pair of the numbers of the same sign, the minimal value is $-1$
I feel like I should apply the same pigeon hold principle for large $n$, but coudln't find a way..
Hint: Let's work on a simpler version, where $P_i = a_i a_{i+1} \in \{-1, 1\}$, $S = \sum P_i$, $n > 2$.
Show that
1) $S \equiv n \pmod{2}$.
2) $S\geq -n$.
3) The number of $P_i = -1$ is even.
4) Lower bound of $S $ is $- n $ when $n$ is even, $-n+2$ when $n$ is odd.
5) These can be achieved, hence are the minimum (greatest lower bound).
Now, generalize to $P_i =$ product of $4k+2$ terms.