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### Graph Without Crossing Lines

```
Date: 7/19/96 at 11:41:42
From: SURGICON LTD
Subject: Graph Without Crossing Lines

Dear Dr. Math,

there are three houses           1        2           3
and
there are three utilities       gas	    water      electric

How can you connect the three houses individually to the three

(You have to draw a line from each house to each of the utilities
without the lines ever crossing.)

Thanks,
Tone
```

```
Date: 8/15/96 at 10:51:46
From: Doctor Patrick
Subject: Re: Graph Without Crossing Lines

Hi, sorry to tell you this, but that problem is impossible on most
normal surfaces.  The only one I know of where it will work is what is
called a torus--which is basically a doughnut-shaped figure with a
hole in the center.

If you can, try the problem again on a surface like that and see what
happens.  Since a torus might be hard to come by, you can cut a hole
in the center of the paper and allow lines to go onto both sides and
through the hole onto the other side.

Why do you think it works in this shape?

-Doctor Patrick,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics

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