Problem of the Week

Problem of the Week #258 

At a U.N. committee, 25 representatives sit around a circular table. Once an hour there is a ”yes” or ”no” vote on an issue, and each representative behaves as follows: on the nth vote, if his response is the same as the response of at least one of his two immediate neighbours, then he will respond in the same way on the (n + 1)th vote as on the nth vote. But if his response is different from that of both his neighbours on the nth vote, then his response on the (n + 1)th vote will be different from his response on the nth vote. 

Prove that, however everybody responded on the first vote, there will be a time after which no one’s response will ever change. 

Submit your solution to NH 278.