> 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.