I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the T piece. This parity is found with no other piece in the game.
Background: The Tetris playing field has width 10. Rotation is allowed, so there are then exactly 7 unique pieces, each of which is composed of 4 blocks.
For convenience, we can name each piece by a letter. See this Wikipedia page for the Image (I is for the stick piece, O for the square, and S,T,Z,L,J are the others)
There are 2 sets of 2 pieces which are mirrors of each other, namely L,J and S,Z whereas the other three are symmetric I,O,T
Language: If a row is completely full, that row disappears. We call it a perfect clear if no blocks remain in the playing field. Since the blocks are size 4, and the playing field has width 10, the number of blocks for a perfect clear must always be a multiple of 5.
My Question: I noticed while playing that the T piece is particularly special. It seems that it has some sort of parity which no other piece has. Specifically:
Conjecture: If we have played some number of pieces, and we have a perfect clear, then the number of T pieces used must be even. Moreover, the T piece is the only piece with this property.
I have verified the second part; all of the other pieces can give a perfect clear with either an odd or an even number used. However, I am not sure how to prove the first part. I think that assigning some kind of invariant to the pieces must be the right way to go, but I am not sure.
My colleague, Ido Segev, pointed out that there is a problem with most of the elegant proofs here – Tetris is not just a problem of tiling a rectangle.
Below is his proof that the conjecture is, in fact, false.