Tiling a Square with T-, L-, and a Square Tetrominoes

Both L- and T-tetrominoes tile a chessboard. It takes 16 tiles of each kind. The question I wish to ask is whether

It is known that a straight tromino (or even three of them) could not be replaced with square tiles. What is happening with T- and L-tetromino tilings?

Solution

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The answer is, No.

In the common coloring of a chessboard, a T-tetromino covers an odd number of white (1 or 3) and an odd number (also, 1 or 3) of dark squares. It follows that an odd number (15 in particular) will cover an odd number of white and an odd number of dark squares. A square tile would add 2 white and 2 dark squares to such a tiling, leaving the totals odd. But the number of white and dark squares on the board is even, implying that no such tiling is possible.

1. Anany and Maria Levitin, Algorithmic Puzzles (Oxford University Press, 2011), #33(f)

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