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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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Some Thoughts Lower limit for N dimensions | Comment 5 of 10 |

A lower limit for N dimensions is found by a generalization of armando's suggestion: 2N+4.

This is described by any two distinct regular N-dimensional polygon formed by faces of equilateral triangles with edge length 1. So for N=2, this is an equilateral triangle, for N=3 a triangular pyramid, etc.

For any three points, two of three of the points have to be on the same polygon, and each point in each polygon is a distance of 1 from each other point in the polygon. So this solution gives us a lower bound for the answer in N dimensions: for two dimensions, this is 6, for 3 dimensions, this is 8.

  Posted by Avin on 2005-05-24 01:18:00
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