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Discussion of the Tetris Applet

Why do we care that we can't win this Tetris game? We always lose every Tetris game eventually.

Most Tetris games end because the player makes a sequence of mistakes that let the board fill up. If a player were perfect, it's possible that they could play forever. Several Tetris fans have attempted to write computer programs to play the game and met with little success.

The pattern recognition and strategies most Tetris players rely on are difficult to program into a computer. However, it might be theoretically possible to create a perfect computer Tetris player which could successfully play Tetris until its plug was pulled. The arguments presented on this page prove that that is impossible.

Why can't you win a Tetris game in which only alternating Z- and S- shaped pieces appear? There are two main reasons -- Z and S pieces are skinny on the ends and fat in the middle, and the board is 2n blocks wide, where n is an odd number (n=5).

By "skinny on the ends and fat in the middle" I mean that if you rotate a Z or S piece so that it is only two rows high, it will consist of one block on the left, one on the right, and two in the middle. If you set it down on your accumulated stack of Tetris pieces in this orientation, you will add one, two, and one block to the different rows of the stack. If a row is deleted from the board, you will still have one more block where the middle of the Tetris piece landed than you did before the piece was placed.

Because my special Tetris game provides only Z and S pieces, that discrepancy in the height of the stack can never be remedied -- no matter what you do the middle rows will always have more blocks in them than the outer ones. Over time, these inequities will accumulate if pieces continue to be laid down horizontally, and the game must end.

A similar argument shows that if a Z or S is rotated to be two columns wide (and three rows high), it must be placed in one of five two row wide "lanes" that evenly divide the board. If it lands straddling two lanes, it will again contribute to an accumulation of pieces on the stack that cannot be remedied.

Why is it important that there are five lanes on the board, and that five is an odd number? You can see from the statements I made above that the only possible way to continue this special Tetris game indefinitely is to stack the pieces head to tail in the lanes. An experienced Tetris player will quickly realize that unless S's are stacked on other S's and Z's are stacked with Z's, holes form on the board that can't be erased by stacking the pieces in their lanes.

However, because there is an odd number of lanes, there must always be an unequal number of S's and Z's at the top of the stacks. If you continue to place S's atop S's and Z's atop Z's, this inequality will make some of the stacks grow faster than others. Eventually, you will be forced to place a Z on an S stack, an S on a Z stack, or lose the game.

Since stacking the pieces in lanes will force us to create "holes" in the stack of pieces on the board that will eventually bring the game to a close, we see that there is no strategy by which we can continue this Tetris game indefinitely -- we must eventually lose.

If we return to the topic of our Tetris playing computer program, probability says that allmost every Tetris game will contain a long string of alternating S's and Z's. Hence, even a perfect Tetris player will lose almost every game he, she, or it plays.

For more details and the proofs of these statements, see the paper "How to Lose at Tetris" in the Mathematical Gazette, July 1997.

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Created: 1/7/96
Copyright © 1995 by The Geometry Center. All rights reserved.