Imagine that a painter went down to a mathematical plane and colored all of the points on that plane one of three colors.
Prove that there exist two points on this plane, exactly one meter apart, that have the same color.
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Submitted by Jonathan Waltz
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Rating: 4.1000 (10 votes)
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Solution:
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First, draw an equilateral triangle. Each of the vertices must be a different color, red, green and blue. Then, using the line connecting green and blue, make another equilateral triangle with the point furtherest away from the original red dot having to be red, because blue and green are already used.
Then, imagine that the paint was still wet, and you swung the whole diamond shape around, pivoting it around the red dot from the original triangle, the red dot staying in one place. Now you have a red circle around the outside. Since the circles diameter is greater than one meter, There has to be somewhere on that circle a chord to connect two of the points on the circle that is exactly one meter long. Then there are two red points exactly one meter apart. |
