Problem
COM-B1-M07-P015 Seventeen Vertical Dominoes
#15
★★★☆☆ Level 3 of 5
Can an \(8\times8\) board be tiled by dominoes so that exactly \(17\) dominoes are vertical?
Color the columns alternately in two colors.
Color the columns alternately. A horizontal domino always covers one cell of each column color. A vertical domino covers two cells of one color. Since the board has equal color counts, vertical dominoes must be split equally between the two types of columns. Their number must be even. Since \(17\) is odd, such a tiling is impossible.
Same idea on a full board with more pieces.