BUTTON TRAP ROOM

You are trapped in a small phone booth shaped room. In the middle of each side of the room there is a hole. In each hole there is a push button that can be in either an off or on setting. You can't see in the holes but you can reach your hands in them and push the buttons. You can't tell by feel whether they are in the on or off position. You may stick your hands in any two holes at the same time and push neither, either, or both of the buttons as you please. Nothing will happen until you remove both hands from the holes. You succeed if you get all the buttons into the same position, after which time you will immediately be released from the room. Unless you escape, after removing your hands the room will spin around, disorienting you so you can't tell which side is which. How can you escape?

Now generalize. You are in a room with N sides, each side having a hole with a push button. What is the minimum number of hands you need to escape the trap room?

Note: Regarding the original setup with N = 4, the fewest possible turns that I know of is seven

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