Eventually constant votes. Let's say that there are $2n+1$ persons sitting at a round table. Each time these people will vote by yes or no. The process starts by an initial vote and in the next vote if at least of person number $i$'s neighbors voted the same vote as theirs, person $i$ keeps his vote in the next voting. If both neighbors of person $i$ voted opposite to them he has to change his vote in the next voting.
Here's an example where $n=1$ I'll denote The vote of person number $i$ in the $k$th turn by $P_i^k= Y$ or $N$. So let's say $$P_1^1=Y, P_2^1=N, P_3^1=Y$$
Then $$P_1^2=Y
, P_2^2=Y, P_3^2=Y$$
Here  $P_2$ changed his vote because both of their neighbors voted opposite to him.
$$P_1^3=Y, P_2^3=Y, P_3^3=Y$$
And So on, so eventually no one will change their vote.
I think that the key to solving this problem is by keeping track of a type of persons along the voting. The people who both of their neighbors have opposite votes to them. If we can somehow prove that the number of these people will shrink to $0$ we're done.
 A: Some observations that may point you in the right direction:

*

*once a person's vote is 'stable', it can't change regardless of neighbors' status

*there can be at most 2 rounds of votes

How can we make these observations concrete? Let's start by noting that if we are not in stationary form, there must be some sub-chain of the people sitting around the table that is not in stationary form. That is, we can't have the entire table oscillating. Why? This is a consequence of the fact that there are an odd number of people around the table. So at some position around the table, there must be two people who are adjacent and share the same vote. This pair will be stable and convert adjacent neighbors. Let's call any stable block S, and WLOG assume that the stable block consists of two Yes voters.
For example, if we have n = 2, 5 voters, say the initial configuration is YNYNY, the 1st and 5th positions are stable, which converts the rest of the unstable part (seats 2-4) into Yes votes. So for n = 2 we have SNYN. It is clear to see that this subchain will become stable in the next round.
The main idea here is to find a good notation for expressing that we turn our initial configuration into two components: unstable subchains, and stable blocks. By  the same argument as before, the neighbors adjacent to the stable blocks will have their behavior determined by the stable blocks, which determines the behavior of the neighbors adjacent to them, and so on. So for any configuration, we have noticed that there will be some stable part, which converts the surrounding unstable part into a stable part in the next round. Can you find a way to make this argument rigorous?
