Problem
COM-B1-M07-P023 Exactly Half Vertical
An \(8\times8\) board is tiled by dominoes. Prove that if the number of vertical dominoes is odd, such a tiling is impossible. Then give an example with exactly \(16\) vertical dominoes.
For impossibility, use column coloring. For the example, split the board into two halves.
Color the columns alternately. As before, horizontal dominoes cover one cell of each color, while vertical dominoes cover two cells of one color. Since the whole board has equal color counts, vertical dominoes must be split equally between the two types of columns. Hence their total number is even. Thus an odd number of vertical dominoes is impossible. Example with \(16\): tile the first four columns entirely by vertical dominoes, giving \(16\) vertical dominoes; tile the remaining four columns by horizontal dominoes along rows.
Includes both an impossibility proof and a construction.