At games night at UW, the mathematician friends of mine have a particularly quick and clever way to determine who starts each game which they call "Shoots" (don't ask me why...) Anyone who has done any studies with modular arithmetic or group theory should be quite comfortable with it.
Okay, so here's how it goes: one player calls "Shoots!" and on three, everyone reveals a random number of fingers, rock-papers-scissors style. Now add up all the fingers and count that many around the circle of players clockwise starting with the player who called for "Shoots". Ta-da. Whoever you finish with is the start player.
Of course, in math terms, if x is the total number of fingers and the players are numbered clockwise as 1,...,n where 1 is the caller of "Shoots" then the start player is simply x modulo n. See, easy!
Start Player deck of cards. But I think math is WWWAAAAYYYYY more fun.....