Domino Tiling on Altered Chessboard

If we take dominoes, each of which can cover up two of these squares, can we perfectly tile this chessboard after removing two squares from opposite corners?

This problem presents a fascinating exploration of the concept of parity and the use of graph-based thinking in discrete mathematics to solve tiling puzzles. The challenge focuses on the ability to cover a modified chessboard, missing two squares at opposite corners, using standard domino pieces, which each cover exactly two squares. The key mathematical abstraction in this problem is considering the coloring of a standard chessboard, which alternates black and white squares. Upon removing two squares from opposite corners, observe that these removed squares are of the same color, disrupting the uniform parity of total colors on the board. Dominoes cover one black and one white square, thus requiring an equal number of both colors to achieve a perfect tiling.

This parity mismatch implies that a perfect tiling is impossible, since there will be more of one color than the other, leading to unmatched squares. The strategy to uncovering a solution to this kind of problem extends beyond physical trial and error and into understanding fundamental principles governing configurations. In particular, you can model this scenario as a bipartite graph-based problem, where nodes represent individual squares, and edges symbolize the dominoes' coverage capacity. By examining this structural representation, the problem inherently reveals its own constraints, demonstrating the power of abstraction in revealing solvability or impossibility within seemingly simple puzzles.