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# Combinatorics #3

A $$4\times19$$ rectangle is composed of unit squares, each colored either red, green or blue. Prove that there exists a rectangle whose sides are parallel to the sides of the $$4\times19$$ rectangle formed by connecting the centers of $$4$$ squares of the same color.

Note by Victor Loh
2 years, 11 months ago

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Nice problem!

First you have to understand how such a rectangle can be formed: by having two $$4\times 1$$ rectangles inwhich two of their squares are colored with the same colors and they are in the same positions.

Looking at the $$4\times 1$$ rectangles, since $$4$$ squares are colored with $$3$$ colors, by PHP(short for pigeonhole principle from now on :) ) there exists two squares with the same color, we will call these two squares "linked".

Since there are $$19$$ of these rectangles, by PHP there exists $$\lceil \frac {19}{3} \rceil=7$$ rectangles inwhich the colors of their "linked" squares are the same. Note that there are $${4\choose 2}=6$$ possible ways for the "linked" squares to position, Hence by PHP there exists two rectangles inwhich their "linked" squares are in the same positions and we are done. · 2 years, 11 months ago

Keep up the good work! I'll be posting more interesting problems (Around Level 3 perhaps) · 2 years, 11 months ago