Using your Head is Permitted

With three squares, however, one already has two possibilities: the straight-line triomino and the corner triomino (a.k.a. "the corner").

Consider tiling a 64x64 square by corner triominoes (composed of three 1x1 squares each). Clearly, 64x64 does not divide by three, so it is impossible to tile the entire 64x64 square. One square must be left untiled. But which one?

Specifically: consider all possible tilings where corner triominoes cover all the 64x64 square except one 1x1 square. For which 1x1 squares are there such tilings where they are the only untiled square? For which squares are there no such tilings? Give a complete characterization and prove your answer.

List of solvers: