Problems

You have an \(n\times m\) chocolate bar. You break the bar into two pieces along a line between its squares, then your friend and you take turns (your friend starts) choosing one of the pieces and breaking it again along a line between its squares. The player who cannot make a move loses. For which values of \(n\) and \(m\) do you win?

We are going to play the following game: look at the grid below, where each cell is painted black or white

choose any two cells that are next to each other and switch both of their colours. By switch, we mean: if a cell is black, change it to white, and if it is white, change it to black. Is it possible to turn the entire grid black?