After numerous attempts and several different approaches, I concluded that the game is impossible to beat, and a quick Google search reveals that I was correct.
The only way to "win", is to cheat using a feature of the Flash player. If you can get to the point where you have everything connected except one wire (which shouldn't be that difficult), drag the last wire until it's almost touching either a house or a utility, then right-click to bring up the Flash menu, then move your mouse through the house/utility, then left click again on the other side. This causes your wire to be drawn underneath the house/utility, allowing you to connect everything.
What a lame solution. It would have been much better if there was a proper way to solve this puzzle.
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Another addicting Flash puzzle
