A Damn Good Puzzle (Oops)
The Great Censor has, in his inscrutable wisdom, laid down a judgment on every finite string of letters in our alphabet--each is either allowed or forbidden.
Say the petty censors intercept the infinite string of symbols ababababab... .
The Great Censor doesn't mind a little arbitrariness in our choice of parsing--he is one arbitrary dude himself. However, he expects our judgments to be resounding; he doesn't like it if a parsed text alternates back and forth between allowed and forbidden words.
Achieving this has been a matter of great vexation among the censors. However, one day a clever young apprentice censor, still far from internalizing the entire system of judgments, had a realization:
No matter how the set of finite strings is divided between allowed and forbidden by the Great Censor, and no matter what infinite string of letters a petty censor is presented with, it will always be possible to parse the infinite string in such a way that the parsed words are either all allowed or all forbidden--with the possible exception of the first word, which may disagree with the others.
So why is this true?