44. Vasya placed numbers from 1 to $101^{2}$ once each in all the cells of a $101 \times 101$ board. Petya chooses a cell on the board, places a token on it, and wants to make as many moves as possible so that the number under the token keeps increasing. In one move, Petya can move the token to any cell within a $5 \times 5$ square centered on the cell where the token is currently located (the token must remain within the board). What is the maximum number of moves Petya can definitely make, regardless of how Vasya arranges the numbers?

(S. Berlov)
44. Answer: 8 moves (you can visit 9 cells).

![](https://cdn.mathpix.com/cropped/2024_05_06_e84b7905ec5809a7a2b4g-1.jpg?height=228&width=277&top_left_y=760&top_left_x=1689)

In any $3 \times 3$ square, regardless of the arrangement of numbers, all cells can be visited, which gives us 8 moves. We will color the board in 9 colors such that the colors correspond to pairs of remainders from dividing the coordinates of the cell by 3. Then, the chip must change color with each move (the distance between different cells of the same color along any coordinate axis is divisible by 3). We will place the numbers in ascending order, starting from 1, as follows: first on the cells of the first color, then on the cells of the second color, and so on, finally ending on the cells of the ninth color. Then the chip cannot make more than 8 moves, as there is no way to move from a cell of the ninth color.