3. Two players take turns drawing lines on a plane, with no line being drawn twice. The player who, after their move, causes the number of pieces the plane is divided into by the drawn lines to be divisible by 5 for the first time, wins. Who will win with correct play: the player who moves first, or their opponent, and how should they play to win?
Answer. Second. First solution. The second player, on their first move, draws a line parallel to the one drawn by the first. If the first player, on their second move, draws a line parallel to the two already drawn, after their move, the plane will be divided into 4 parts. Then the second player draws a line parallel to the three already drawn, and wins: the number of parts becomes exactly 5. If, however, the first player, on their second move, draws a line intersecting the two already drawn, after their move, the plane will be divided into 6 parts. Then the second player draws a line intersecting the three already drawn at three different points, and also wins because the number of parts becomes 10. Second solution. The second player, on their first move, draws a line intersecting the one drawn by the first. If the first player, on their second move, draws a line parallel to one of the two already drawn or passing through the point of their intersection, after their move, the plane will be divided into 6 parts. Then the second player draws a line intersecting the three already drawn at three different points, and wins because the number of parts becomes 10. If, however, the first player, on their second move, draws a line intersecting both already drawn at two different points, the plane will be divided into 7 parts. Then the second player draws a line parallel to one of the drawn lines and not passing through their points of intersection, and the number of parts again becomes 10.