We number the squares of an $8 \times 8$ chessboard with numbers from 1 to 64, and we call the "distance" between two adjacent squares the absolute value of the difference between the numbers on them. (We consider two squares to be adjacent if they share a vertex.) What is the largest number such that a greater distance between adjacent squares never occurs in any numbering of the chessboard?


Consider any numbering of the chessboard, and place a chess piece - specifically the king, which can move to adjacent squares in one step - on the square that received the number 1. No matter where the king is, it can reach any square on the chessboard, including the one numbered 64, in at most 7 moves. The total distance traveled along the way is at least 63, so the king must have moved at least "9" at least once, since $7 \cdot 8$ is only 56. This means that no matter how we number the chessboard, there is a pair of adjacent squares whose distance is at least 9, so the sought number is at least 8.

However, it is easy to see that the sought number is exactly 8. If we number the chessboard "continuously," that is, we write the number $(n-1) \cdot 8 + k$ on the $k$-th square of the $n$-th row, then the distance between adjacent squares can be 1, 8, 7, or 9. Therefore, there is a numbering in which no distance greater than 9 appears between adjacent squares.