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 $\color{purple}{\text{T}}$ 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 ($\color{cyan}{\text{I}}$ is for the stick piece, $\color{goldenrod}{\text{O}}$ for the square, and $\color{green}{\text{S}},\color{purple}{\text{T}},\color{red}{\text{Z}},\color{orange}{\text{L}},\color{blue}{\text{J}}$ are the others)

There are $2$ sets of $2$ pieces which are mirrors of each other, namely $\color{orange}{\text{L}}, \color{blue}{\text{J}}$ and $\color{green}{\text{S}},\color{red}{\text{Z}}$ whereas the other three are symmetric $\color{cyan}{\text{I}},\color{goldenrod}{\text{O}}, \color{purple}{\text{T}}$

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 $10$, the number of blocks for a perfect clear must always be a multiple of $5$.

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

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.


