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Topic: impossible problem
Replies: 3   Last Post: Jan 20, 2004 9:25 AM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: impossible problem
Posted: Jan 19, 2004 12:18 PM
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On Sun, 18 Jan 2004, John Berglund wrote:

> It is impossible unless you bend the rules
(like allowing one of the lines to pass underneath a house.)
>
> Here is the simplest way of explaining this that I know:

Call the houses A,B,C. Call the utilities X,Y,Z.

[I now simplify John's explanation.]

Then the lines A--Z, Z--B, B--X, X--C, C--Y, Y--A form a
closed circuit, which can be distorted so as to look like a hexagon:-

A--Z
/ \
Y B
\ /
C--X .
But now we must supplement this by the three diameters A--X, B--Y, C--Z
each of which must either lie entirely inside the hexagon or entirely
outside. But we can have at most one lying inside (for if two, they
must cross), and similarly at most one lying outside - a contradition.

John Conway





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