Tile Swap

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You’re tiling a floor with 4×1 and 2×2 tiles when you accidentally break one. A tile of the other shape is available. Show that it’s not possible to cover the floor by rearranging the tiles.

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Color the tiles as shown. Now every 4×1 tile must cover 0 or 2 black squares, and every 2×2 tile must cover one black square. So we can’t successfully exchange one for the other. From Arthur Engel’s excellent Problem-Solving Strategies, 1998.

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Color the tiles as shown. Now every 4×1 tile must cover 0 or 2 black squares, and every 2×2 tile must cover one black square. So we can’t successfully exchange one for the other.

From Arthur Engel’s excellent Problem-Solving Strategies, 1998.

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