Question

Suppose that each point in the coordinate plane is colored either red or blue. Show that there must always be two points of the same color that are exactly one unit apart.

Answer 1

Here's a clue:
Equilateral Triangle

Answer 2

so if the three vertices are colored the same, then that means that there is always two points of the same color exactly one unit apart, right?

Answer 3

thats not exactly right, but you have the right idea about coloring the vertices. To be extremely formal about it, do you know the Pigeon-hole Principle? If not thats fine.

Answer 4

no, what is it?

Answer 5

The pigeonhole principle states:
"If you have m holes, and m+1 pigeons, some hole will have at least 2 pigeons." Not even joking. It sounds really basic, but some of the stuff you can prove with it is incredible. Try googling it later, its really interesting. Anywhos back to your problem. You have 3 vertices, and 2 colors to choose from....

Answer 6

where did you get that joemath ...i never heard of it ...but thanks : )

Answer 7

so two of the vertices have to be the same color!

Answer 8

google it, its probably one of my favorite theorems.
@tulip thats right!

Answer 9

and your done lol

Answer 10

I just googled it. wow it can be applied to so many cases. Anyways thanks for the help! :)

Answer 11

This is probably the hardest problem ive done using the PHP. It just shows how awesome the theorem is:
Each of the given 9 lines cuts a given square into two quadrilaterals whose areas are in proportion 2:3. Prove that at least three of these lines pass through the same point.

Answer 12

well couldn't a line divide the square diagonally to get two quadrilaterals whose areas are in proportion 2:3? and there are four points are a horizontal midline that divides the square in a 2:3 ratio so since there are 9 lines, at least 3 of them have to pass through a single point, right?

Answer 13

yep, thats correct!

Answer 14

that was awesome btw.

Answer 15

thanks :)