Except for $n=1$, the first player wins by breaking off a triangle of side $n-1$ and passing the remaining trapezoidal piece made of $2n-1$ triangles of side $1$ to the second player. From now on, each player can only break off a triangle of side $1$, so after $2n$ steps the second player is facing a single chip piece, hence loses.

mavropnevma wrote: Except for $n=1$, the first player wins by breaking off a triangle of side $n-1$ and passing the remaining trapezoidal piece made of $2n-1$ triangles of side $1$ to the second player. From now on, each player can only break off a triangle of side $1$, so after $2n$ steps the second player is facing a single chip piece, hence loses. I think you misread the last part of the question it says "A player who leaves exactly one chip to the opponent wins" in other word the player who eats the last piece wins. In your case the second player eats the last piece.