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3 colors (Posted on 2003-05-01) Difficulty: 3 of 5
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.

  Submitted by Jonathan Waltz    
Rating: 4.1000 (10 votes)
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSlight variation with Desmos pictureLarry2024-01-19 10:13:50
Possible Solution?Brittany2004-04-17 10:55:10
Wait! I have something else!Ryan2003-05-07 18:38:50
Try this proofRyan2003-05-07 15:46:43
re(2): Proof by contradictionCharlie2003-05-07 06:15:40
re(2): Proof by contradictionBrian Smith2003-05-07 03:40:58
re: Proof by contradictionRyan2003-05-06 19:25:56
re: Elegant proofRyan2003-05-06 19:11:37
SolutionChaz2003-05-02 13:19:00
SolutionElegant proofBryan2003-05-01 13:01:09
re: Proof by contradictionDJ2003-05-01 12:46:00
re: Proof by contradictionCharlie2003-05-01 10:08:13
SolutionProof by contradictionBrian Smith2003-05-01 08:37:44
SolutionProofCharlie2003-05-01 08:13:04
UnproveableDuCk2003-05-01 08:08:40
re: Two Or Three ?friedlinguini2003-05-01 06:02:43
re(3): Could be rightCharlie2003-05-01 05:38:00
re: Three-Dimensional PlaneCharlie2003-05-01 05:32:54
re(2): Could be rightHank2003-05-01 05:29:46
re: Two-Dimensional PlaneCharlie2003-05-01 05:29:07
SolutionThree-Dimensional PlaneRavi Raja2003-05-01 05:27:12
SolutionTwo-Dimensional PlaneRavi Raja2003-05-01 05:25:28
QuestionTwo Or Three ?Ravi Raja2003-05-01 05:24:58
re(3): Could be rightCharlie2003-05-01 05:24:13
re: Could be rightBrian Smith2003-05-01 04:54:27
re(2): Could be rightRavi Raja2003-05-01 04:46:56
re: Could be rightfriedlinguini2003-05-01 04:36:43
Some ThoughtsCould be rightHank2003-05-01 04:16:30
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