True story: the Cambridge Maths Society, The Archimedeans, had a journal called Eureka (of course). It happily published interesting but not important bits of mathematics. One student wrote a proof of an impossible sequence for any width board. It's even impossible if you have perfect knowledge of the sequence. All you need is to drop Ss and Zs in an irrational order.
I think this got published in 1990, but I can find little evidence even if the journal's existence online.
The proof itself is quite self-evident: given a perfectly randomly block generator, and infinite play time, eventually you're going to produce a very long sequence of left- (or right-) facing S-blocks. Since those are impossible to eliminate (there's always one block left open on a row) eventually you lose.
This actually isn't true for modern variants of Tetris. A correct implementation of the random number generator requires that the 7 pieces be shuffled into a bag and then drawn randomly, ensuring an even distribution, and disallowing long runs without a desired piece. (Maximum of 12.) The bag is re-shuffled every 7 blocks in the sequence.
In that case it would probably still be possible, but the proof would have to be
found by example. It seems like an advanced version of the Eight Queens
puzzle: which sequence of bags guarantees a losing board state?
Not that one. It's from 1996, the one I'm thinking of was 1993 latest and I think 1990. There was also a previous article that demonstrated that the player always won for 3-omino Tetris (not a massively interesting result).
As observed elsewhere, neither work according to official Tetris rules.
I misunderstood "board" as blackboard and thought that a Cambridge student had written a proof that, for any arbitrarily wide blackboard, there was a sequence which it was impossible to write on the board.
This seems like the sort of theorem that a college student might be inclined to prove.
I think this got published in 1990, but I can find little evidence even if the journal's existence online.