When I was younger, before I went to school, I tried bring water to a boil, then adding the eggs with a spoon. I would usually loose a third of the eggs to cracking as the cold shells hit the hot water.
Now, I start by placing the eggs in the pan, covering with 1/2" of hot water out of the tap, covering the pan, and placing it on a hot burner. As long as my electric burner is red-hot when I put the pan on, it will boil in a few minutes. I then turn the heat down enough to maintain a simmer, covered, and start a 14-minute timer. After 14 minutes, uncover, drain, fill with enough cold water to cover the eggs, then add a few ice cubes to stop the cooking. Perfect eggs each time.
I see many different times listed, especially "off the heat". What people are doing is poaching the eggs, in the shell. The varying time will result in differently cooked egg yolks, depending on how long you actually boiled the eggs, and your altitude. Altitude only comes into play because the lower air pressure affects the temperature at which water boils. Lower air pressure means a lower boiling point (put a bowl of water into a vaccuum chamber, and remove the air. At one point, room-temperature water will boil, but because you are lowering the air pressure, temperature is also going down, and when it hits 32 degrees, the "boiling" water will actually freeze at the same time). So, longer times "at altitude" are a compensation for the water boiling at a lower temperature.
Also, you have carry-over cooking to worry about. Even if you drain the hot water from the eggs, the eggs themselves are still hot, and are still cooking, until they are immersed into cold water. The faster you stop the cooking, the more chances you have that your eggs will not overcook and get the grey-green ring on the yolk.