I'm wondering if it is possible to make a 3x8 grid using 6 tetrominoes: 2 Is, 2 Ls, a Z skew, and a square tetromino. I believe it may be impossible but would like to know why.
All the ways it can be tiled. Five ways, if you want to search by hand... Two ways use a mirrored pair of Ls. I have a program that does this sort of thing, started writing it when the IBM PC was released so it's out of control now in terms of code length/maintainability.
It is possible; I wrote a small C++ program to find a solution: https://godbolt.org/z/rdjb7n66s
L I I I I L O O L Z Z L L L O O L L Z Z I I I I