It is well-known that there is a knight's tour on an \( 8\times 8 \) chessboard; that is, a path of knight moves starting and ending at the same point which touches each other square exactly once along the way. Let's think about knight's tours on other rectangular chessboards.
(a) Consider the following question: how many knights can you place on an \( m \times n \) chessboard such that no two knights attack each other? Show that the answer is at least \( mn/2 \).
(b) Show that if there is a knight's tour on an \( m \times n \) chessboard, then \( mn \) is even.
(c) Suppose there is a knight's tour on an \( m \times n \) chessboard. Show that there are exactly two ways to place \( mn/2 \) knights on the board so that no two attack each other. (The "exactly" is important here!)
(d) Show that there is no knight's tour on a \( 4 \times n \) chessboard. (Hint: use part (c).)