structural:reformatting
Changing the rules

K 3, 3 {\displaystyle K_{3,3}} is a toroidal graph, which means that it can be embedded without crossings on a torus, a surface of genus one. These embeddings solve versions of the puzzle in which the houses and companies are drawn on a coffee mug or other such surface instead of a flat plane. There is even enough additional freedom on the torus to solve a version of the puzzle with four houses and four utilities. Similarly, if the three utilities puzzle is presented on a sheet of a transparent material, it may be solved after twisting and gluing the sheet to form a Möbius strip.

Another way of changing the rules of the puzzle that would make it solvable, suggested by Henry Dudeney, is to allow utility lines to pass through other houses or utilities than the ones they connect.