Three Houses, Three Utilities

Date: 07/15/99 at 01:43:02
From: Chris
Subject: Lines, etc.

I know that you have answered this before: the question about the 
three houses and the three utilities (gas, electricity, water).  

Well, the guy who gave me this puzzle says there is a way of solving 
it in 2D, without any tricks. He says that it is simple, once you 
figure it out. I don't get it. Everywhere, it says that it can only be 
done using 3 dimensions. Can you solve it using 2 dimensions?  How?

Thank you very much  :)

Date: 07/15/99 at 12:38:34
From: Doctor Rob
Subject: Re: Lines, etc.

Thanks for writing to Ask Dr. Math!

You can only solve this if you allow one of the utility lines to run
through someone else's house, or through one of the other utility
companies, which I suppose is possible, but is usually forbidden by 
the conditions of the puzzle.

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

Date: 07/15/99 at 12:46:43
From: Doctor Peterson
Subject: Re: lines, etc.

Hi, Chris.

He may not call it a trick, but any solution that's really 2D (that 
is, done just by drawing non-intersecting curves on a flat sheet of 
paper) has to twist the rules somehow. He might, for example, draw the 
houses as rectangles and say that it's legal to open the front and 
back doors of one house and pass a pipe through. I call that a trick. 
Another trick is to solve it on the surface of a donut (a torus) and 
point out that any surface is itself 2-dimensional, even though it 
exists in a 3-dimensional space. Or you can allow going around to the 
other side of the paper through a hole, which is essentially the same 
thing, as this answer points out:

  http://mathforum.org/dr.math/problems/tone.7.19.96.html   

When the problem is stated carefully in mathematical terms (continuous 
non-intersecting curves from each of three points to each of three 
other points), there's no solution; but presented in terms of houses 
and utilities (which are inherently three-dimensional), there are lots 
of ways to get around it.

I'd like to hear what his answer is.

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