Knight's tour from all starting positions
The Wikipedia article links to several useful papers. According to the paper Solution of the knight’s Hamiltonian path problem on chessboards, there is no such tour; see Theorem 3.1(ii). They show that a knight's tour on the 5x5 board always starts or ends at a corner. It's also easy to see that both endpoint squares have the same color. Since the (5,4) square has a different color than the corners, no knight's tour starts at (5,4).