KöMaL 2021 K684

$a)$ Sophie and Bertie are playing a game that involves breaking a bar of chocolate into pieces. The chocolate bar consists of $10\times 5$ squares. They take turns in splitting the chocolate along the dividing lines. The player who first breaks off a single square will lose the game. In each move, they are only allowed to touch and split one piece. Sophie starts the game. Can she make sure that she will win, whatever Bertie's moves are? $b)$ After the first game, Bertie wants to strike back. He wants Sophie to start again, but with a different rule: the player who first breaks off a single square will win the game. Can Sophie make sure that she will win again?