184. The King traversed an $8 \times 8$ chessboard, visiting each square exactly once and returning to the starting square on the last move (the King moves according to the usual rules). When his path was drawn, connecting the centers of the squares he sequentially passed through with line segments, a closed broken line without self-intersections was obtained.

a) Provide an example showing that the King could have made exactly 28 moves horizontally and vertically.

b) Prove that he could not have made fewer than 28 such moves.

c) What is the greatest and the least length that the King's path can have if the side of a cell is 1?
184. a) See Fig. 77,6.

b) Let's highlight a border of 28 edge squares on the board. The king visited each of them during his tour. We will number these squares in the order the king visited them. The entire
![](https://cdn.mathpix.com/cropped/2024_05_21_7b8a6b2625543c3dae75g-183.jpg?height=304&width=628&top_left_y=414&top_left_x=268)

Fig. 77

path of the king will be divided into 28 segments: from square 1 to square 2; from square 2 to square 3; ...; from square 28 to square 1. The king's path has no self-intersections. Therefore, the squares 1 and 2, 2 and 3, ..., 27 and 28, 28 and 1 are adjacent squares of the border. (If, for example, squares 1 and 2 are not adjacent, then they divide the border into 2 parts, and the king's path from any square of one of these parts to any square of the other will intersect the segment 1-2, which contradicts the problem's condition.) To move to an adjacent square of the border, the king must, at some step, move from a square of one color to a square of another color, i.e., make a move either vertically or horizontally. Therefore, the king's path contains at least 28 such moves.

c) It is clear that the length of the king's path is no less than 64, and such a short path exists (Fig. 77,6). On the other hand, we have proven that the king's path contains no fewer than 28 segments of length 1. Therefore, the king could make no more than 36 diagonal moves, so the entire path is no longer than \(28 + 36\sqrt{2}\). (An example of such a path is shown in Fig. 77, a).)