# The Mathematics of Tetris

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 $$T\color{purple}{\text{T}}$$ piece. This parity is found with no other piece in the game.

Background: The Tetris playing field has width $$1010$$. Rotation is allowed, so there are then exactly $$77$$ unique pieces, each of which is composed of $$44$$ blocks.

For convenience, we can name each piece by a letter. See this Wikipedia page for the Image ($$I\color{cyan}{\text{I}}$$ is for the stick piece, $$O\color{goldenrod}{\text{O}}$$ for the square, and $$S,T,Z,L,J\color{green}{\text{S}},\color{purple}{\text{T}},\color{red}{\text{Z}},\color{orange}{\text{L}},\color{blue}{\text{J}}$$ are the others)

There are $$22$$ sets of $$22$$ pieces which are mirrors of each other, namely $$L,J\color{orange}{\text{L}}, \color{blue}{\text{J}}$$ and $$S,Z\color{green}{\text{S}},\color{red}{\text{Z}}$$ whereas the other three are symmetric $$I,O,T\color{cyan}{\text{I}},\color{goldenrod}{\text{O}}, \color{purple}{\text{T}}$$

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 $$1010$$, the number of blocks for a perfect clear must always be a multiple of $$55$$.

My Question: I noticed while playing that the $$T\color{purple}{\text{T}}$$ 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 $$T\color{purple}{\text{T}}$$ pieces used must be even. Moreover, the $$T\color{purple}{\text{T}}$$ 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.

Thank you,