The governor's mansion in Frankfort, Kentucky.traveler1116/Getty ImagesForty-five of the 50 states have official mansions, which governors can live in rent-free.Some governors' mansions are open to the public for tours.Arizona, Idaho, Massachusetts, Rhode Island, and Vermont don't have designated mansions.After a long day serving constituents, governors return home to sprawling, historic buildings.They're welcomed by 20-foot entranceways; their …