[math-fun] Games of no strategy

What are fun examples of combinatorial games that (like Conway and Paterson's game of Brussels Sprouts) appear to be games of strategy but whose outcome doesn't depend on what either player does? One of my favorites is Impartial Cutcake, aka the Candy Bar Game. The initial state is a chocolate rectangle, scored into unit squares. On any given turn, a player may take any rectangular piece bigger than a 1-by-1 square and divide it along a horizontal or vertical scoring-line into two smaller rectangular pieces. The two players alternate. When no further divisions are possible, the game ends, and the player who made the last move wins (and, if you like, gets to eat all the pieces). Jim Propp