Question 62
Hugo plays a game: he places a chess piece on the top left square of a $20 \times 20$ chessboard and makes 10 moves with it. On each of these 10 moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $\mathrm{X}$ on the square that the piece occupies. When Hugo plays this game over and over again, what is the largest possible number of squares that could eventually be marked with an X? Prove that your answer is correct.