1. The squares of a $9 \times 10$ chessboard are colored red, green, or blue. Each $3 \times 3$ square on the chessboard contains exactly three red, three green, and three blue squares. What is the maximum number of red squares?
(A) 32
(B) 34
(C) 36
(D) 39
(E) 40
1. The answer is $\mathbf{( C )}$. Consider a $9 \times 9$ square of the chessboard, as shown in the figure. This sub-chessboard can be covered with $\frac{9 \times 9}{9}=9$ squares of $3 \times 3$, each of which contains 3 red squares by hypothesis. The red squares are therefore at most 36, i.e., 27 (the $9 \cdot 3$ red squares in the $9 \times 9$ square) plus 9 (the squares outside the $9 \times 9$ square). On the other hand, the coloring in the figure (where $R, V, B$ stand for red, green, blue) provides an example in which the red squares are exactly 36.

| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |
| $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ | $\mathrm{V}$ | $\mathrm{B}$ | $\mathrm{R}$ |