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.

#Combinatorics

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

Xuming Liang - 6 years, 11 months ago

Keep up the good work! I'll be posting more interesting problems (Around Level 3 perhaps)

Victor Loh - 6 years, 11 months ago

Thanks! I'm trying to improve my solution writing skills for combinatorics problems..

Xuming Liang - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$