Short answer:
It is impossible for a knight to move from the bottom left corner to the top right corner of an 8x8 chessboard, stepping on each grid exactly once.
Explanation:
To move from the bottom left corner to the top right corner, the knight needs to make 63 moves, each to a different square, since it cannot revisit any square. However, a knight's move alternates between two colors on a chessboard. Starting from the bottom left square (which is black), each move the knight makes alternates between black and white squares. Since there are 32 squares of one color and 30 of the other on an 8x8 chessboard, the knight cannot end on a square of the opposite color from where it started. Thus, it's impossible for the knight to reach the top right corner, which is of a different color from the bottom left corner, without repeating any squares. Therefore, the task is impossible.
