Your friend has given you a very unusual birthday present.
It is a round box, and inside the lid are $n$ buttons equally spaced in a circle.
The box functions like this; you can press two buttons, and then the box will shut by itself and spin the buttons around by one place (you don't know whether it spins them clockwise or anticlockwise, though, and it can go a different way each time), before letting you press another two buttons.
Each time you open the box, it looks identical - there's no feedback on which buttons you've pressed before either, they're springy and they remove fingerprints and markings and they all reset to the same temperature and they yada yada, don't post a non-theoretical answer please :)
Once all buttons have been pressed at least once, it will sing you an amusing rendition of happy birthday that you really want to hear!
For what n do you have a strategy to ensure that the box will eventually sing happy birthday?