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### 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.
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Puzzles
Middle School Geometry
Middle School Puzzles
Middle School Two-Dimensional Geometry

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