65. At the beginning of the game, Little Boy and Karlson have one piece of chocolate in the form of a $2019 \times 2019$ square grid. Each move, Little Boy divides some piece along the grid lines into three rectangular pieces, and Karlson eats one of these three pieces at his choice. The game ends when no more moves can be made. If the total number of moves made is even, Little Boy wins; if it is odd, Karlson wins. Who wins with correct play?

( D. Shiryayev)
65. Answer: Karlson will win. Let's call a piece of chocolate big if it can be cut, and small if it cannot. Initially, there is only one big piece, and at the end of the game, there are 0. Karlson can play in such a way that the parity of the number of big pieces after his move will definitely change: one big piece is destroyed by Little Man's move, after which 0 to 3 new big pieces may appear. If the number of new big pieces is odd, then there is definitely one and Karlson will eat it. If the number of big pieces is even, then there is at least one small piece and Karlson will eat exactly the small piece. Thus, the number of big pieces after each pair of moves changes parity, and all moves must change the parity of the number of big pieces, so the number of moves is odd.