8.70 In chess, the king traverses an $8 \times 8$ chessboard, visiting each square exactly once and returning to the starting point (the king's moves follow the usual rules of chess), and when the centers of the squares the king has visited are connected by line segments in sequence, the resulting path is a non-intersecting closed polyline.
(1) Provide an example showing that the king can move a total of exactly 28 steps along horizontal and vertical lines.
(2) Prove that the total number of steps the king makes along horizontal and vertical directions is at least 28.
(3) If the side length of each square is 1, find the maximum and minimum possible lengths of the king's path.
[Solution] (1) See the figure (a) below.
(2) Consider the boundary of the chessboard composed of 28 edge squares. Clearly, the King passes through each of these boundary squares exactly once as it traverses the entire chessboard. Number these edge squares in the order the King passes through them as $1, 2, \cdots, 28$, and divide the King's entire path into 28 segments: from square 1 to square 2, from square 2 to square 3, ..., from square 28 to square 1. According to the given conditions, the King's path does not intersect itself, so square $i$ and $i+1$ must be adjacent, for $i=1, 2, \cdots, 28$ (square 29 is square 1). Otherwise, if square 1 and 2 are not adjacent, they would divide the boundary into two parts. When the King moves from a square in one part to a square in the other part, the path taken must intersect the path from square 1 to square 2, which is a contradiction. Since squares $i$ and $i+1$ are adjacent, they must be of different colors. Therefore, when the King moves from square $i$ to square $i+1$ after several steps, at least one of these steps must be from one color to another, and this step must be either along a horizontal or vertical line. This means that the total number of steps the King takes along horizontal and vertical lines is at least 28.
(3) Clearly, the total length of the King's path is no less than 64, and there exists a path of length 64 (as shown in figure (b)). On the other hand, from (1) and (2), we know that at least 28 steps of the King's entire path are along horizontal or vertical lines. Therefore, the maximum possible length of the path is $28 + 36\sqrt{2}$.