Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:
Determine whether it is possible to tile a standard $8 \times 8$ chessboard with $15$ $L$ -tiles and $1$ $T$ -tile of the shapes below:

(Each tile covers four squares of the chessboard. The tiles can be flipped and rotated at will.)

Solution

Solution:
Let the squares of the chessboard be colored black and white in the usual way. Note that the $L$ -tile always covers two squares of each color, while the $T$ -tile covers either $1$ black and $3$ white or $3$ black and $1$ white squares. Since the chessboard has equal numbers of black and white squares, it is impossible to cover it with these tiles.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.