2006 Romanian NMO Problems/Grade 7/Problem 2

Problem

A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).

Solution

For $n\leq6$, consider this coloring for a 6x6 board:

\[\begin{tabular}{|c|c|c|c|c|c|} \hline R&Y&G&R&Y&G \\ \hline G&R&Y&G&R&Y \\ \hline Y&G&R&Y&G&R \\ \hline R&Y&G&R&Y&G \\ \hline G&R&Y&G&R&Y \\ \hline Y&G&R&Y&G&R \\ \hline \end{tabular}\]

We can take the top $n$-by-$n$ grid of this board as a coloring not satisfying the conditions. For $n\geq7$, we note that each row or column must have at least one color with 3 or more squares by the pigeonhole principle, so our answer is 7.

See also

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