Maena and Théodore are playing a game. They play on a square grid consisting of $99 \times 99$ cells. Two cells are considered adjacent if they share a vertex or a side.
Initially, Maéna numbers the cells of the grid from 1 to $99^{2}$, in an arbitrary manner. Théodore then places a token on one of the cells of the square, and he allows himself moves of the following form: he can move the token from one cell to another only if these cells are adjacent and if the new cell on which the token lands has a strictly greater number than the old cell.
What is the minimum number of moves Théodore can guarantee, regardless of how Maena placed her integers?
First, Théodore can always manage to make at least three moves. To do this, he just needs to select a $2 \times 2$ square inside the $99 \times 99$ square, then traverse its four cells, which are necessarily adjacent since they share a vertex.
Conversely, here is how Maena can proceed to prevent Théodore from making more than three moves. She numbers the rows and columns from 1 to 99, then groups the $99 \times 99$ cells of the square into four categories:
$\triangleright$ Category 1 contains the cells located in an odd row and an odd column;
$\triangleright$ Category 2 contains the cells located in an odd row and an even column;
$\triangleright$ Category 3 contains the cells located in an even row and an odd column;
$\triangleright$ Category 4 contains the cells located in an even row and an even column.
Two cells of the same category are never adjacent.
Next, Maena places her integers, in ascending order, in cells of category 1, then category 2, then category 3, and finally category 4. Thus, Théodore can never move between two cells of the same category, and even less decrease the category. Any move therefore increases the category of the cell in which Théodore's token is located, thus limiting him to a maximum of three moves.

Comment from the graders In a problem like this, there are two things to do:

1. Propose a strategy that Maena can use to prevent Théodore from making too many moves;
2. Propose a strategy that Théodore can use to make at least $k$ moves (with $k$ to be chosen) regardless of Maena's strategy.
Among the students who looked for a strategy for Maena, many concluded that she could prevent Théodore from making more than three moves. Among those who looked for a strategy for Théodore, many concluded that he could always make at least three moves.
Unfortunately, few students thought to look for strategies for both Maena and Théodore. In particular, proving that Théodore can always make at least three moves when Maena follows the strategy she chose is almost useless, since Maena still has $\left(99^{2}\right)!-1 \approx 10^{34864}$ alternative ways to number the grid, and it remains to prove that Théodore can also make at least three moves in each of these $\left(99^{2}\right)!-1$ cases.