Difference between revisions of "2006 Romanian NMO Problems/Grade 7/Problem 2"
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A square of side <math>n</math> is formed from <math>n^2</math> unit squares, each colored in red, yellow or green. Find minimal <math>n</math>, 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). | A square of side <math>n</math> is formed from <math>n^2</math> unit squares, each colored in red, yellow or green. Find minimal <math>n</math>, 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== | ==Solution== | ||
| − | + | For <math>nleq6</math>, consider this coloring for a 6x6 board: | |
| + | |||
| + | RYGRYG | ||
| + | GRYGRY | ||
| + | YGRYGR | ||
| + | RYGRYG | ||
| + | GRYGRY | ||
| + | YGRYGR | ||
| + | |||
| + | We can take the top <math>n</math>-by-<math>n</math> grid of this board as a coloring not satisfying the conditions. | ||
| + | For <math>n\geq7</math>, 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== | ==See also== | ||
*[[2006 Romanian NMO Problems]] | *[[2006 Romanian NMO Problems]] | ||
[[Category:Olympiad Combinatorics Problems]] | [[Category:Olympiad Combinatorics Problems]] | ||
Revision as of 01:46, 18 March 2009
Problem
A square of side 
is formed from 
unit squares, each colored in red, yellow or green. Find minimal , 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 
, consider this coloring for a 6x6 board:
RYGRYG GRYGRY YGRYGR RYGRYG GRYGRY YGRYGR
We can take the top -by- grid of this board as a coloring not satisfying the conditions.
For 
, 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.
