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Zimmerman's Family Points (Posted on 2005-05-23) Difficulty: 5 of 5
What is the maximum number of points in the Euclidean plane with the property that given any three points, at least two are at distance one apart?

No Solution Yet Submitted by owl    
Rating: 4.4000 (5 votes)

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Bryan's solution? | Comment 4 of 10 |

Take a regular pentagon with side lengths 1 in the xy-plane.  Add points on both sides of this plane so that each is distance 1 from the vertices of the regular pentagon.

For any 3 points on the regular pentagon, at least two are adjacent, whence they are at distance 1.  If one of the 3 points is off the pentagon, then at least one of the 3 is on the pentagon, whence those two are at distance 1.  So the maximum number of points is at least 7, as Bryan suggests.


  Posted by McWorter on 2005-05-24 00:42:24
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