Question: Prove that any sequence of the word “buffalo” of length n>1 is a valid English language sentence.
First, let n be odd. We start with n=3: “Buffalo buffalo buffalo”; that is, some buffalo do buffalo buffalo, i.e., some buffalo are buffaloed by buffalo. But of course the buffalo who are buffaloing may themselves be buffaloed by buffalo, so just as some cats that watch mice are chased by dogs, or as we say, cats dogs chase watch mice, buffalo that buffalo buffalo themselves buffalo buffalo, and we can say that buffalo buffalo buffalo buffalo buffalo. Anytime we have the noun buffalo, we can add the relative clause “who are buffaloed by buffalo”, or better, instead of the noun phrase “buffalo who are buffaloed by buffalo”, we may say simply “buffalo that buffalo buffalo”, then add the rest of the sentence, yielding “Buffalo that buffalo buffalo buffalo buffalo”, or even better, “Buffalo buffalo buffalo buffalo buffalo”. To a sentence consisting of n (odd) occurrences of the word, we can produce a sentence of n+2 occurrences.
But just as a dog that chases cats is a dog that chases, buffalo that buffalo some buffalo are buffalo that buffalo, so from one of our sequences of an odd number of occurrences, we can lop off the final direct object, producing a sequence of an even number of occurrences that is a grammatical sentence. For any n>1, odd or even, a sequence of n occurrences of “buffalo” is a grammatical English sentence!