Competition problem — Combinatorics

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Combinatorics Difficulty 5.4 AIME, harder Find the answer Italy

Problem:
The cells of a $9 \times 10$ chessboard are colored red, green, or blue. Every $3 \times 3$ square of the chessboard contains exactly three red cells, three green cells, and three blue cells. At most how many red cells can there be?

Pick one

Solution

Solution:
The answer is $\mathbf{( C )}$ . Consider a $9 \times 9$ square of the chessboard, as in the figure. This sub-chessboard can be covered with $\frac{9 \times 9}{9}=9$ $3 \times 3$ squares, each of which by hypothesis contains 3 red cells. The red cells are therefore at most equal to 36, that is 27 (the $9 \cdot 3$ red cells in the $9 \times 9$ square) plus 9 (the cells 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 cells 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: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.