Hamming distance for conjugates

Let x , y be strings of equal length. The Hamming distance h ( x , y ) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y , we say x and y are conjugates. We consider f ( x , y ) , the Hamming distance between the conjugates x y and y x . Over a binary alphabet f ( x , y ) is always even, and must satisfy a further technical condition. By contrast, over an alphabet of size 3 or greater, f ( x , y ) can take any value between 0 and | x | + | y | , except 1; furthermore, we can always assume that the smaller string has only one type of letter.