$(5$ points$)$ Tetris ${?}^{TM}$ is a game where you rotate and place one$-$sided $4-$polyominoes, also called one$-$

sided Tetrominoes or simply Tetrominoes $($defined below$),$ in various positions trying to clear lines to

score points while praying to the Tetris mg gods. A Tetromino is defined as an object built from four

equal sized squares such that all squares are connected and each square shares a common side with

another square. It is known that, up to some integer number of $90$ degree rotations, the Tetrominoes

can be classified as one of the following typest:

For completion of our definition of Tetromino does not include things like:

Suppose that you wanted to expand Tetris ${?}^{TM}$ to allow for more block types according to the rules

above. Use case analysis to prove that up to some integer number of $90$ degree rotations,

all Tetrominoes are all of one of the above types.

Hint $1$: You can use case analysis where the maximum vertical extent of the Tetromino is $1,2,3$

and $4.$ Here by "maximum vertical extent" I mean the maximum number of blocks that are placed

directly on top of each other under the particular orientation you are considering. i$.$e$.$ I$-$block as

pictured above has a maximum vertical extent of $4$ but when rotated by $90$ or $-90$ the vertical extent

is $1,$ the T$-$block as pictured is $2$ but when rotated by $90$ or $-90$ the vertical extent is $3$ etc.

Hint $2$: If you follow hint $1$ then in some cases you will have subcases. Try to minimize these subcases

by using the rotational symmetry of Tetrominoes.

Hint $3$: The worst case analysis approach to this problem $($that I can imagine a student trying$)$ would

require $1820$ cases but this problem can be solved in $15$ cases or less $-$ Use hints $1$ and $2.$