Is it possible to tile a $10\times10$ chessboard with (non-overlapping) T-tetrominos?
If so, how? If not, prove it's impossible.
Bonus: Which Tetris pieces can used to tile a 10$\times$10 board, allowing reflections?
It is certainly posssible to tile a regular $8\times 8$ chessboard. Below is a failed attempt to extend that tiling to a $10\times 10$ board, which misses some squares in the upper left and lower right. The colors don't matter, they're just there to make the picture more clear.
$\qquad\qquad\qquad\qquad$
Answer
It is not possible. The area of a $10 \times 10$ checkerboard is $100$, so it takes $25$ T pieces to have the same area. The checkerboard has the same number of red and black squares, but each piece covers three of one color and one of the other. $25$ pieces cannot cover $50$ squares of each color, the most even they can get is $51-49$
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