Kimoo A.V. The King walked around the chessboard, visiting each square exactly once and returning to the starting square on the last move. (The King moves according to the usual rules: in one move, he can move horizontally, vertically, or diagonally to any adjacent square.) When his path was drawn, sequentially connecting the centers of the squares he visited, a closed broken line without self-intersections was obtained. What is the smallest and the largest length that it can have? (The side of a cell is equal to one unit.) |  |  |

#
The King made 64 moves. Since the length of each move is either 1 (a straight move) or $\sqrt{2}$ (a diagonal move), the total length of the path is certainly no less than 64. A path of length 64 is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_de7bb834a1e75b584fa6g-21.jpg?height=706&width=723&top_left_y=1305&top_left_x=656)

We will show that the length of the King's path cannot be greater than $28+36 \sqrt{2}$. Consider two consecutive exits $A$ and $B$ of the King to the edge of the board. If these boundary squares are not adjacent, then the segment of the King's path between them divides the board into two parts, each containing whole squares. However, in this case, the King cannot move from one part to the other, which contradicts the condition. Therefore, the squares $A$ and $B$ must be adjacent and, consequently, of different colors. Since the color of the square does not change during diagonal moves, there must be a straight move between any two consecutive exits to the edge. Since there are 28 boundary squares, there are also 28 exits to the edge, and thus there must be at least 28 straight moves. The following figure shows that exactly 28 "straight moves" can be used.

![](https://cdn.mathpix.com/cropped/2024_05_06_de7bb834a1e75b584fa6g-22.jpg?height=692&width=694&top_left_y=11&top_left_x=682)

## Answer

The minimum length is 64, and the maximum is $28+36 \sqrt{2}$.

Submit a comment