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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 cottages and companies are drawn on a [[coffee mug]] or other such surface instead of a flat plane can be solved.{{r|parker}} Similarly, if the 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 Dudeney, is to allow utility lines to pass through
==Properties of the utility graph==
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