Sorry for revive old post

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An easier solution to a):

(Oops bad internet made the post double)

Alternative argument: A $3\times n$ table can be covered if $n~$ is even, but not if $n~$ is odd. The proof is that if you pick a corner and cover it with one V-tromino, then you always get existence of a space that can only be covered by a V-tromino uniquely, and you inevitably end up reducing the problem to the covering of a $3\times (n-2)$. As a result, you eventually wind up with either $n=0$ (problem solved), or $n=1$ (trivially unsolvable).