witzar 18 Report post Posted July 2, 2013 Each point of the plane is painted either red or blue. Prove, that there exists a rectangle with all vertices of the same color. Share this post Link to post Share on other sites

0 vistaptb 2 Report post Posted July 2, 2013 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. Share this post Link to post Share on other sites

0 Anza Power 9 Report post Posted July 2, 2013 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 Share this post Link to post Share on other sites

0 Yoruichi-san 116 Report post Posted July 3, 2013 (edited) 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 July 3, 2013 by Yoruichi-san Share this post Link to post Share on other sites

0 Anza Power 9 Report post Posted July 4, 2013 *headdesk*just realized it's rectangles and not triangles... Oh well, bonus for everyone else: prove there is one triangle too... Share this post Link to post Share on other sites

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

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

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