Coloring the Checkerboard
We take a board (say a rectangle divided into unit squares) and color it like a checkerboard: black and white alternately. Then we ask:
What happens to the coloring when we place certain tiles?
The Key Insight
The Key Insight
A domino (1×2 or 2×1) always covers one black and one white square.
A 1×1 square covers only one color (black or white).
An L-tromino covers 2 of one color + 1 of the other.
A square covers two of both colors
This gives a color-balance invariant:
If every tile you use covers equal numbers of black and white, then any region you tile must itself have equal numbers of black and white squares.
If the region has imbalance (like 2 extra blacks), tiling is impossible.