This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
The game of cigarettes is played as follows:
Two players take turns placing a cigarette on a circular table. The cigarettes
can be placed upright (on end) or lying flat, but not so that it touches any
other cigarette on the table. This continues until one person loses by not
having a valid position on the table to place a cigarette.
Is there a way for either of the players to guarantee a win?
The first person wins by placing a cigarette at the center of the table,
and then placing each of his cigarettes in a position symmetric (with
respect to the center) to the place the second player just moved. If the
second player could move, then symmetrically, so can the first player.