Henry Ernest Dudeney/Modern Puzzles/156 - Water, Gas and Electricity/Solution

by : $156$

 * Water, Gas and Electricity

Solution
There are some tricksy solutions which involve running the pipes underneath the actual houses, such as the below, which apply to the letter of the law:


 * Dudeney-Modern-Puzzles-156-solution.png

but the fact is that if you treat the houses and pipes as vertices and edges, the problem has no solution.

Proof
The graph as defined is the complete bipartite graph $K_{3, 3}$.

The problem is equivalent to attempting to prove that $K_{3, 3}$ is a planar graph.

However, from Kuratowski's Theorem, $K_{3, 3}$ is non-planar.