Chessboard Domino Dilemma

This puzzle presents a classic domino-tiling question on a standard chessboard, with specific squares removed. In the first scenario, two opposite corners are excluded and the candidate must decide whether the remaining pattern of squares can be perfectly tiled by dominoes, each covering two orthogonally adjacent cells. The twist is that the excluded cells are fixed in a particular geometric relationship. The variation asks the same feasibility question but now the two removed squares are constrained only by their colors, one from each color class. This style of brain teaser often appears in quantitative interviews to see how candidates move from a concrete picture to an abstract structural property.

Solving it leans on invariants and parity-style arguments rather than brute-force tiling. An interviewer expects the candidate to notice the underlying coloring structure of the board and connect it to how each domino must be placed. Strong answers articulate why some configurations are impossible in principle, rather than trying to "fit pieces." They also clearly separate the two cases, checking the necessary conditions for a tiling and reasoning carefully about when those conditions are sufficient.

What it tests

Domino tiling problems on checkerboard-patterned boards are governed by the invariant that each domino always covers one square of each color. This means that for a perfect tiling, the board must contain an equal number of black and white squares after any removals. If the numbers are unequal, tiling is impossible, because each domino reduces both counts by one. This principle holds because the coloring partitions the board into two interlocking sets, and dominoes always bridge the sets, so any imbalance cannot be corrected by placement. The parity and color-balance constraints are fundamental and apply regardless of board size or domino orientation.