Competition problem — Combinatorics

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Combinatorics Difficulty 5.4 AIME, harder Find the answer

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?

Pick one

Solution

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.

$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$
$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$	$\mathrm{V}$	$\mathrm{B}$	$\mathrm{R}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.