Consider the $9 \times 9$ board shown below. The rows are numbered from 1 to 9.

![](https://cdn.mathpix.com/cropped/2024_05_01_874784cb479b0f26034ag-1.jpg?height=406&width=454&top_left_y=962&top_left_x=641)

Figure 116.1

We color the cells of the odd-numbered rows of the board with the colors blue and white, alternately, starting with blue, and we paint the cells of the even-numbered rows of the board gray and red, alternately, starting with the color gray.

(a) How many cells were painted with each color?

(b) What is the maximum number of pieces of the form

| $\square \square$ that can |
| :--- | :--- |

be placed, without overlapping, on this board?
(a) Each odd row contains 5 blue houses and 4 white houses. Since the board has 5 odd rows, the number of blue houses is $5 \times 5 = 25$ and the number of white houses is $5 \times 4 = 20$.

Similarly, each even row has 5 gray houses and 4 red houses, and the board has 4 even rows. Thus, the number of gray houses is $4 \times 5 = 20$ and the number of red houses is $4 \times 4 = 16$.

(b) No matter how we place the piece

![](https://cdn.mathpix.com/cropped/2024_05_01_874784cb479b0f26034ag-1.jpg?height=100&width=146&top_left_y=2337&top_left_x=995)
it will always cover one house of each color on the board. Since the board has only 16 red houses, the number of pieces must be less than or equal to 16.

Figure 116.2 shows a configuration with exactly 16 pieces.

Hint: For part (b), check how many houses of each color are covered when placing a piece on the board.

![](https://cdn.mathpix.com/cropped/2024_05_01_874784cb479b0f26034ag-1.jpg?height=468&width=486&top_left_y=1425&top_left_x=1433)

Figure 116.2

## Related Problem

Is it possible to divide an $8 \times 9$ board into $1 \times 6$ rectangles?

Hint: See the problem Numbers on a $4 \times 4$ Board, at level 1, on page 100.