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The three utilities problem of connecting three houses to three utilities by nine non-crossing paths can be formalized as finding a drawing of the graph $K_{3,3}$ as a planar graph (no such drawing exists, so it is an impossible puzzle). But that formalization hides a lot of assumptions. What surface are the lines on? Can lines pass through other houses or utilities? Answering these questions differently can make the puzzle solvable. https://en.wikipedia.org/wiki/Three_utilities_problem, now a Good Article on Wikipedia.