|image1=3_utilities_problem_torus.svg|caption1=Solution on a torus
|image2=3 utilities problem moebius.svg|caption2=Solution on a Möbius strip}}
K<submath>K_{3,3}</submath> is a [[toroidal graph]], which means that it can be embedded without crossings on a [[torus]], a surface of genus one,.{{r|harary}} andThese thatembeddings 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 can be solved.{{r|parker}} AThere is even enough additional freedom on the torus to solve a version of the puzzle with four houses and four utilities on the torus can also be solved.{{r|obeirne|early}} 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]].{{r|larsen}}
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.{{r|dud17}}
