Question 32
Twenty-five men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the $n^{\text{th}}$ vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{th}$ vote as on the $n^{th}$ vote; but if his response is different from that of both his neighbours on the $n$-th vote, then his response on the $(n+1)$-th vote will be different from his response on the $n^{th}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.