How many $6 \times 1$ dominoes can be placed on a 9 x 10 chessboard without overlapping? (The dominoes can be placed "parallel" to the sides of the board.)
We will show that at most 14 dominoes can be placed on the board.

First, we prove that 15 dominoes cannot be placed.

Assume that 15 dominoes can indeed be placed. Since the area of the board is \(10 \cdot 9 = 6 \cdot 15\), which is equal to the total area of the dominoes, this can only be true if the dominoes cover the entire board without gaps.

![](https://cdn.mathpix.com/cropped/2024_05_02_f7ad5e2cdadb764966aeg-1.jpg?height=360&width=395&top_left_y=346&top_left_x=846)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_05_02_f7ad5e2cdadb764966aeg-1.jpg?height=360&width=390&top_left_y=850&top_left_x=846)

Figure 2

Color the squares of the board white and black as shown in Figure 1. It is easy to verify that each domino covers three white and three black squares. However, the board has 48 black and 42 white squares, so the dominoes cannot cover the entire board.

In Figure 2, we show an arrangement where 14 dominoes are on the board.

Therefore, 14 dominoes can be placed on the board, but 15 cannot.

Note. The fact that 15 dominoes cannot be placed can also be proven using other colorings. Here are some examples (Figures 3-5).

![](https://cdn.mathpix.com/cropped/2024_05_02_f7ad5e2cdadb764966aeg-1.jpg?height=358&width=395&top_left_y=1656&top_left_x=846)

Figure 3

![](https://cdn.mathpix.com/cropped/2024_05_02_f7ad5e2cdadb764966aeg-1.jpg?height=404&width=393&top_left_y=2156&top_left_x=844)

![](https://cdn.mathpix.com/cropped/2024_05_02_f7ad5e2cdadb764966aeg-2.jpg?height=339&width=398&top_left_y=123&top_left_x=842)

Figure 5