RU-ALLRUSSIAN 1978 Problem 11

Two players play a game by moving a piece on an n x n chessboard. The piece is initially in a corner square. Each player may move the piece to any adjacent square (which shares a side with its current square), except that the piece may never occupy the same square twice. The first player who is unable to move loses. Show that for even n the first player can always win, and for odd n the second player can always win. Who wins if the piece is initially on a square adjacent to the corner?