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|image1=3_utilities_problem_torus.svg|caption1=Solution on a torus
|image2=3 utilities problem moebius.svg|caption2=Solution on a Möbius strip}}
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
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 the cottages or utilities.{{r|dud17}}
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