You are faced with numerous lightbulbs, with many cables running amongst them, each cable connecting two lightbulbs together. In addition, each lightbulb has a button, and pressing it changes the state (i.e. turns it on if it was off, and vice versa) of the lightbulb, and all the neighboring lightbulbs (all these connected to it).
Provided that all the lights are turned off in the beginning, prove that you can light them all up!