Tile Swap

Futility Closet presents a classic mathematical puzzle: imagine you're tiling a floor with four-by-one and two-by-two tiles when you accidentally break one. A tile of the other shape is available. Can you rearrange them to cover the floor? The answer is no, and here's why. If you color the tiles in a specific pattern, every four-by-one tile must cover either zero or two black squares, while every two-by-two tile covers exactly one. That mathematical constraint makes it impossible to swap one shape for the other and complete the job. The puzzle comes from Arthur Engel's Problem-Solving Strategies from nineteen ninety-eight.

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