  A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In
1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over
3 symbols. We consider the online variant of this result in which a
nonrepetitive sequence is constructed during a play between two players: Bob is
choosing a position in a sequence and Alice is inserting a symbol on that
position taken from a fixed set A. The goal of Bob is to force Alice to create
a repetition, and if he succeeds, then the game stops. The goal of Alice is
naturally to avoid that and thereby to construct a nonrepetitive sequence of
any given length. We prove that Alice has a strategy to play arbitrarily long
provided the size of the set A is at least 12. This is the online version of
the Theorem of Thue. The proof is based on nonrepetitive colorings of
outerplanar graphs. On the other hand, one can prove that even over 4 symbols
Alice has no chance to play for too long. The minimum size of the set of
symbols needed for the online version of Thue's theorem remains unknown.
