Figuring out the shortest possible chess game is easy: we’re all familiar with the Fool’s Mate, a checkmate by Black that comes after just two moves (1. f3 e5 2. g4 Qh4# being the basic idea). See also here. Much more interesting are questions about the longest chess games played.
If you’ve been playing tournament chess for a while, chances are that you’ve broken the 100-move barrier at least once, and many have probably played games much longer than that (I personally know I’ve played a game of over 130 moves, …