Let $n\ge 2$ be a natural number. We consider the following game. Two players write a sequence of $0$'s and $1$'s. They start with an empty line and alternate their moves. In each move, a player writes $0$ or $1$ at the end of the current sequence. A player loses if his digit completes a block of n consecutive digits that repeats itself in the sequence for the second time (two occurrences of the block may overlap).
For instance for $n=4$ a sequence produced by such a game would look as follows : $00100001101011110011$ ( the second player lost the last move because $0011$ is repeated)
Show that for $n=4$ ,the first player has a winning strategy.
For $n$ odd I have a winning strategy for the second player. The strategy is whatever the first player writes the second player writes the complement of it. But for $n=4$, I have no idea how to solve it. Please help.