Help with Colouring Proof for tiling a 3×9 board with L-Dominos

Question

I define an L-Domino as a 2×2 square with one piece removed. I want to prove that it is not possible to tile a 3×9 board with L-Dominos. My initial thought is to tile the board with 3 colours such that each L-Domino covers each colour once. But this isn't actually possible, since there are ways to tile the board so that not all colours are on each tile. Instead then, I was thinking I could say using the colouring below it must be the case that adjacent tiles must always contain the same number of each colour. Namely, 2 of each colour. But we can't use an even number of tiles since $3 \cdot 9=27$ which is the number of total tiles we need to cover. But then I'm wondering that argument would be a bit too handwavy. Is there a better colouring to use for this problem? I also used the chess colouring but got stuck with that approach too.

The colouring I'm considering would be as follows:

$
XYZXYZXYZ \\
YZXYZXYZX \\
ZXYZXYZXY$

Any hints to solve this problem would be much appreciated!

Answer 1

Use this two-coloring, inspired by the linear programming dual variables:

ABABABABA
BBBBBBBBB
ABABABABA

Each A requires two Bs, but there are 10 As and only 17 Bs.

Alternatively, each A requires a different tile, but there are 10 As and only 9 tiles.