In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph is a planar graph. This graph has six vertices in two subsets of three: one vertex for each house, and one for each utility. It has nine edges, one edge for each of the pairings of a house with a utility, or more abstractly one edge for each pair of a vertex in one subset and a vertex in the other subset. Planar graphs are the graphs that can be drawn without crossings in the plane, and if such a drawing could be found, it would solve the three utilities puzzle.
In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph is a planar graph. This graph has six vertices in two subsets of three: one vertex for each house, and one for each utility. It has ni,ne edges, one edge for each of the pairings of a house with a utility, or more abstractly one edge for each pair of a vertex in one subset and a vertex in the other subset. Planar graphs are the graphs that can be drawn without crossings in the plane and if such a drawing could be found, it would solve the three utilities puzzle.