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# Rectangle on red-blue plane

## Question

Each point of the plane is painted either red or blue.

Prove, that there exists a rectangle with all vertices of the same color.

## Recommended Posts

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Well,you don't really need all of the plane, just a descrete number of points:

try to paint all the vertices of all of the small triangles and you'll see you'll always end up with one of just one color

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Consider the points (x, 1), (x, 2), and (x, 3). If (a, 1), (a, 2), and (a, 3) have the same color combination as the x-points, a rectangle is formed because two of the x-points must be the same color. If these points are, say, (x, 1) and (x, 3), then the rectangle is between (x, 1), (a, 1), (x, 3), and (a, 3). Since there are an infinite number of x-values and only a finite number of color combinations for those three points, there must be two x-values with this same color combination and thus a rectangle.

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Nice, vista, efficient and simple solution...I was trying to draw grids of right triangles

(Just thought it was about time someone actually said something nice about someone else's solution ;P)

Edit: Oh, and thanks for the puzzles, btw, witzar, they've been fun and interesting .

Edited by Yoruichi-san
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*headdesk*just realized it's rectangles and not triangles...

Oh well, bonus for everyone else: prove there is one triangle too...

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