Problem
COM-B1-M07-P020 Is the Two-Cell Claim True?
On an \(8\times8\) board, dominoes cover all cells except two. Is it necessarily true that the two uncovered cells have the same color?
How many cells of each color do dominoes cover?
The full \(8\times8\) board has \(32\) cells of each color. Dominoes cover equal numbers of black and white cells. If one uncovered cell were black and the other white, the covered cells would include \(31\) black and \(31\) white cells, which is consistent. Thus the proposed statement is false: the two uncovered cells may have different colors. For example, leave two adjacent cells and tile the rest. Therefore the correct conclusion is that coloring does not force the two cells to have the same color.
Counterexample is important: the method should not become magic.