Question 17
Juku and Miku play the following game on a grid of dimensions $n \times m$: In the beginning, all unit squares are white. Each player on their turn paints one white unit square either red or blue of their choice, but no two unit squares with a common side or a common vertex can be painted the same color. The players take turns and Juku starts. A player who cannot make the allowed move has lost. Is it possible for Juku to win the game regardless of how Miku plays if:
a) $n = 2023$ and $m = 2023$;
b) $n = 2023$ and $m = 2024$;
c) $n = 2024$ and $m = 2024$?
a) $n = 2023$ and $m = 2023$;
b) $n = 2023$ and $m = 2024$;
c) $n = 2024$ and $m = 2024$?