Question 174
Two players, Cat and Mouse, play the following game on a $4 \times 4$ checkerboard. Each player places a checker on a cell of the board (Cat goes first). Then, the two players take turns moving their checkers to an adjacent square, either vertically or horizontally (Cat again goes first). If, after either player's move, the two checkers occupy the same square, Cat wins. Otherwise, if each checker has made 2013 moves without this happening, Mouse wins. Determine, with proof, which player has a winning strategy.