Tiling squares with L-Trominoes

Question

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?

Answer 1

Here's a proof of a stronger statement: the $L$-tromino tiles any rectangle with dimensions $3a\times b$ if $a\ge 2, b\ne 1,3$.

Note that the L tromino certainly tiles a $3a\times 2$ rectangle (stack $3\times 2$ rectangles on each other), so if we can show this for $b=5$ we'll be done (since all positive integers other than $1$ and $3$ are a sum of $2$s and $5$s).

To do this when $b=5$, we combine two rectangles together as needed: the $6\times 5$ and the $9\times 5$. They can both be tiled, as seen:

                                           

Stacking as necessary, we obtain $3a\times 5$ rectangles for all $a\ge 2$, which completes the proof. The corollary $b=3a$ is immediate.