# 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
3 years, 11 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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.

- 3 years, 11 months ago

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

- 3 years, 11 months ago

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

- 3 years, 11 months ago