Arrange 7 Points in a Plane...

Date: 10/05/98 at 10:28:09
From: Steve Lesko
Subject: A problem you will never get in 3*10e99 years

Arrange 7 points in a plane so that if any three are chosen, at least 
2 of them will be a unit distance apart.  

I got this problem from the USA Math Talent Search, which is sponsored 
by the NSA, so there must be an answer.

Date: 10/05/98 at 12:01:17
From: Doctor Rob
Subject: Re: A problem you will never get in 3*10e99 years

There must be more to this problem than this.  Arrange them in a row
with 2 units between each adjacent pair.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/05/98 at 19:05:31
From: Anonymous
Subject: Re: A problem you will never get in 3*10e99 years

You have to be able to pick any 3 and have 2 of the points a unit 
apart. The closest I have come is with 6 points, and it's a square 
ABCD, with equilateral triangles ABE and CDF, and F and E are a unit 
apart, but there isnt a 7th point that can be added and have it 
fulfill the requirements.  

I didn't understand the problem at first either.

Date: 10/08/98 at 10:58:37
From: Doctor Rob
Subject: Re: A problem you will never get in 3*10e99 years

Draw a rhombus made up of two equilateral triangles joined together 
along a side. Now draw a second one. They contain eight points, so 
take one of the sharp points of each rhombus and make them coincide.  
You get a figure like this:

         A o
          / \
         /   \
        o-----o
         \   /
          \ /
         B o
          / \
         /   \
        o-----o
         \   /
          \ /
         C o

where all the lines have length 1 unit. The only triple of points that
doesn't include a pair 1 unit apart are {A,B,C}.  

Now rotate the lower rhombus about point B until points A and C are 
exactly one unit apart.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/08/98 at 21:16:26
From: Anonymous
Subject: Re: A problem you will never get in 3*10e99 years

THANK YOU so much for that answer.