Answers 361outside (say in the position indicated by one of the black crosses) the gas can
never reach you without crossing a pipe. Whereas if you put the house in-
side one of the enclosures (as indicated by the stars), then you must be cut
off either from the water or the electricity-one or the other. But the house
must be either inside or outside. Therefore a position is impossible in which
it can be supplied from all three stations without one pipe crossing another.
I hope that is thoroughly convincing. Build your two houses wherever you
like and you will find that those conditions that I have described will always
obtain.
[In modem graph theory this is called the "utilities problem," and the
nonplanar graph that supplies the desired connections is known as a Thomsen
graph. The graph cannot be drawn on the plane without one line crossing an-
other; can it be drawn, without such a crossing, on the surface of a
doughnut?-M. G. J
- CROSSING THE LINES
Let us suppose that we cross the lines by bridges, represented in Figure I
by the little parallels. Now, in Figure 2, I transform the diagram, reducing the
5./\. A~
i-BJE