Is there a finite sequence of pieces of Tetris such that for every way of play you always lose?

I assume you mean a "standard set" of Tetris pieces? It is simple to come up with a piece such that if that piece is repeated you will always lose. – Paul Feb 05 '15 at 19:58

1Also, what are the dimensions of the Tetris board used? If such a sequence exists, it is likely to be dependent on these dimensions. – Paul Feb 05 '15 at 19:59

1It's worth noting that modern versions of Tetris don't draw pieces randomly; it hands out a random permutation of each of the 7 pieces before creating a new permutation, so you'll at most see 12 other pieces before seeing the same piece again. Then there's also the "hold" mechanic which lets you save a piece for later. – Doval Feb 05 '15 at 20:44

Wait, so is it or isn't it true that the line piece is drawn with decreased probability? – asmeurer Feb 05 '15 at 21:29
3 Answers
A game of Tetris (on a 10×20 grid) consisting of alternating S and Z tetrominoes will necessarily end before 70000 tetrominoes are played.
http://euclid.trentu.ca/aejm/V4N1/Tsuruda.V4N1.pdf — Lemma 2.
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Not anymore.
Older versions of Tetris had a uniform stateless randomizer. This would allow a repeating SZsequence that eventually causes a loss. This is possible yet unfathomably unlikely in practice because a competent player can survive this sequence for 150 Tetriminos on a standard 10x20cell matrix, and the odds of an ideal randomizer producing this sequence are 1 in (7/2)^{150} = 4×10^{81}, roughly one in the number of atoms in the observable universe.
But since 2001, the randomizer rule in Tetris Worlds, Tetris DS, and other authentic Tetris games has been the "bag" rule, called "Random Generator" by Tetris developer Blue Planet Software. It deals a permutation of the seven distinct Tetriminos (5,040 possibilities), then reshuffles, then deals another permutation, etc. An SZsequence longer than length 4 is forbidden because a sequence of length 4 has to be sandwiched between two permutations of IJLOT.
It turns out that if you keep a pile of S, T, and Z in columns 14 of a 10x20cell matrix, and J, L, and O in columns 710, you can enter a pattern that repeats every 140 Tetriminos and thus extend the game indefinitely. In 2007, Colour Thief released a proof of this, titled "Playing forever". The following tiling of a 10x56cell rectangle should help you grasp the pattern used by the proof.
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If the board has odd width (any length is fine) then yes, there is trivially such a sequence. Just assume all pieces are the $2 \times 2$ squares. So maybe you should insist that the board width is even?
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2The standard board has even dimensions. I believe it's 10x20, but I could be wrong. – A. Thomas Yerger Feb 05 '15 at 20:43