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K<sub>3,3</sub> is a [[toroidal graph]], which means it can be embedded without crossings on a [[torus]], a surface of genus one,{{r|harary}} and that 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}} 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}}
==Properties of the utility graph==
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