Water, gas, and electricity

The classical mathematical puzzle known as water, gas, and electricity, the (three) utilities problem, or sometimes the three cottage problem, can be stated as follows:

: Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Using a third dimension or going through a company or cottage are illegal. Is there a way to do so without any of the lines crossing each other?

Solution

:: $K\_\{3,3\}$

$n=6$, $m=9$

There is no correct solution; it is impossible to connect the three cottages with the three different utilities without at least one of the connections crossing another.

The problem is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K_3,3 is planar. This graph is often referred to as the utility graph in reference to the problem. [ [http://mathworld.wolfram.com/UtilityGraph.html Utility Graph] from "mathworld.wolfram.com"] The graph is equivalent to the circulant graph Ci₆(1,3). Kazimierz Kuratowski proved in 1930 that K_3,3 is nonplanar, and thus that the problem has no solution. The solution can be adduced as a consequence of the Jordan curve theorem. The complete statement on the planarity of graphs (the Kuratowski reduction theorem) encompasses this result.

But K_3,3 is toroidal, which means it can be embedded on the torus. In terms of the three cottage problem this means the problem can be solved by punching two holes through the plane (or the sphere) and attaching them with a tube. This changes the topological properties of the surface and using the tube we can connect the three cottages without crossing lines. An equivalent statement is that the graph genus of the utility graph is one, and therefore it cannot be embedded in a surface of genus less than one. A surface of genus one is equivalent to a torus.

One cannot redraw graphs with non-zero genus in the plane without edge intersections.

References

External links

* [http://mathworld.wolfram.com/UtilityGraph.html Utility Graph] at MathWorld
* [http://www.cut-the-knot.org/do_you_know/3Utilities.shtml 3 Utilities Puzzle] at cut-the-knot
* [http://www.jimloy.com/puzz/puzzle2.htm Proof That the Impossible Puzzle is Impossible]