On the squares of an $n \times n$ chessboard, there are hedgehogs on each square. Every second, the hedgehogs move to an adjacent diagonal square. What is the minimum number of squares occupied at all times?
Coloring 1: in a checkerboard pattern, we deduce that hedgehogs on one color cannot go to the other, so there are at least two squares.

Coloring 2: we color entire rows alternately, with red rows and blue rows. A hedgehog on a red row will go to a blue row and vice versa.

Thus, a hedgehog on a black and blue square cannot be on the same square as a hedgehog on a black and red square, and the same applies to hedgehogs on white and blue squares and white and red squares. Therefore, there are at least four squares. These four squares can be reached by choosing four squares forming a square and bringing the hedgehogs to one of the squares in the square, alternating the square each turn that each hedgehog is on while remaining within the square: a hedgehog in the square can no longer leave it. We continue until all the hedgehogs are in the square.